2 Affine invariance

In this section, we will present the proof of Theorem 1.1. The following proposition relates integration on a flag manifold to integration on nested Grassmannians (see [Reference Schneider and Weil51, Theorem 7.1.1 on p. 267] for such a result for flags of elements consisting of two subspaces). Since we will use this fact many times throughout this paper, we include the proof. For a subspace $F \subset \mathbb {R}^n$ , we denote by $G_{F,i}$ the Grassmannian of all i-dimensional subspaces contained in F.

Proposition 2.1 Let $ 1\leq r \leq n-1$ and $ \mathbf {r} := ( i_{1}, \ldots , i_{r})$ be an increasing sequence of integers between $1$ and $n-1$ . For $G \in L^1(F^n_{\mathbf {r}})$ ,

(18) $$ \begin{align} \int\limits_{F^n_{\mathbf{r}}} G (\xi_{\mathbf{r}}) d\xi_{\mathbf{r}} = \int\limits_{G_{n,i_{r}} } \int\limits_{G_{F_{r}, i_{r-1}}} \ldots \int\limits_{G_{ F_{2}}, i_{1}} G ( F_{1}, \ldots, F_{r-1}, F_{r}) \, d F_{1} \ldots d F_{r-1} dF_{r}. \end{align} $$

For simplicity, we have suppressed the notation to write $dF_1$ rather than $d\mu _{G_{F_2,i_1}}(F_1)$ , which is the Haar probability measure on the Grassmannian of all $i_1$ -dimensional subspaces of the ambient space $F_2$ ; similarly for all other indices. This convention will be used throughout.

Proof Fix $i_j$ . Denote by $SO(F_{j})$ the subgroup of $SO_n$ acting transitively on $G_{F_{j},{i_{j-1}}}$ . For example, if $F_o=\mathop{\mathrm{span}}\{e_1, \ldots , e_{i_j}\}$ and $E_o=\mathop{\mathrm{span}}\{e_1, \ldots , e_{i_{j-1}}\}$ , then elements of $SO(F_o)$ are given by

$$\begin{align*}\left( \begin{array}{cc} SO_{i_j} & 0 \\ 0 & I_{n-i_j} \end{array} \right) .\end{align*}$$

And, the stabilizer of $E_o$ in $SO(F_o)$ is

$$\begin{align*}\left( \begin{array}{ccc} SO_{i_{j-1}} & 0 & 0 \\ 0 & SO_{i_j - i_{j-1}} & 0 \\ 0 & 0 & I_{n-i_j} \end{array} \right) .\end{align*}$$

The measure $\mu _{G_{F_j},i_{j-1}}$ is invariant under $SO(F_{j})$ . Further, for $g\in SO_n$ and a Borel subset $A \subset G_{F_j,i_{j-1}}$ , we have $ \mu _{G_{g F_j,i_{j-1}}} (gA) = \mu _{G_{F_j,i_{j-1}}} (A)$ .

We will show that both integrals are invariant under the action of $SO_n$ . Fix $g\in SO_n$ . We start with the integral on the right-hand side of (18):

$$ \begin{align*} \int_{G_{n,i_r} } & \int_{G_{ F_r, i_{r-1} }} \ldots \int_{G_{ F_2,i_1 }} G (g^{-1} \cdot(F_{1}, \ldots, F_{r})) \, d F_1\ldots dF_{r-1} dF_r \\ &= \int_{G_{n,i_r} } \int_{G_{ g F_r, i_{r-1} }} \ldots \int_{G_{ g F_2, i_1 }} G (F_{1}, \ldots, F_{r}) \, d (g F_1)\ldots d (g F_{r-1}) d (g F_r) \\ &= \int_{G_{n,i_r} } \int_{G_{g F_r, i_{r-1} }} \ldots \int_{G_{gF_3,i_2}}\int_{G_{ F_2, i_1 }} G (F_{1}, \ldots, F_{r}) \, dF_1 d(gF_2)\ldots d (g F_{r-1}) d (g F_r), \end{align*} $$

where we have sent $(F_{1}, \ldots , F_{r}) \to g \cdot (F_{1}, \ldots , F_{r})$ and then used the invariance property

$$ \begin{align*}\mu_{G_{g F_j, i_{j-1}}} (gA) = \mu_{G_{F_j, i_{j-1}}} (A).\end{align*} $$

Continuing this way for all inner integrals and using the $SO_n$ -invariance of the measure $\mu _{G_{n,i_r}}$ for the outer integral, the above expression reduces to

$$ \begin{align*} \int_{G_{n,i_r} } \int_{G_{ F_r, i_{r-1} }} \ldots \int_{G_{ F_2, i_1 }} G (F_{1}, \ldots, F_{r}) \, d F_1\ldots d F_{r-1} d F_r. \end{align*} $$

Note that at each step $(F_{1}, \ldots , F_{r})$ remains an element of $F^n_{\mathbf {r}}$ , this is to say that the inclusion relation is preserved. The invariance of the integral on the left-hand side of (18) is a consequence of the $SO_n$ -invariance of the measure $\mu _{F^n_{\mathbf {r}}}$ . The proposition now follows by the uniqueness of the $SO_n$ -invariant probability measure on $F^n_{\mathbf {r}}$ (see, for example, Section 13.3 in [Reference Schneider and Weil51]).

The following fact allows one to view an integral of a function on a partial flag as an integral over the full flag manifold. In this case, to avoid confusion, the subspaces of flag manifolds are indexed by their dimension.

Proposition 2.2 Let $ 1\leq r \leq n-1$ and $ \mathbf {r} := ( i_{1}, \ldots , i_{r})$ be an increasing sequence of integers between $1$ and $n-1$ . For a function $ G $ on the partial flag $F_{\mathbf {r}}^{n}$ , denote by $\widetilde {G}$ its trivial extension to the full flag manifold $F^{n}$ , i.e., $\widetilde {G}(F_1, \ldots , F_{n-1}) := G(F_{i_1}, \ldots , F_{i_r})$ . Then

(19) $$ \begin{align} \int_{F^{n}} \widetilde{G}( \eta) d \eta = \int_{F_{\mathbf{r}}^{n}} G(\xi_{\mathbf{r}}) d \xi_{\mathbf{r}}. \end{align} $$

Proof We “integrate out” the Grassmannians that do not contain subspaces that G depends on by repeatedly using the identity

$$ \begin{align*}\int_{G_{ F_{j+1}, j}} \int_{G_{ F_j, {j-1}}} f(F_{j-1}) d F_{j-1} d F_{j} = \int_{G_{ F_{j+1}, j-1}} f(F_{j-1}) dF_{j-1}.\end{align*} $$

On the right-hand side, we integrate over the set of all $(j-1)$ -dimensional subspaces in the ambient $(j+1)$ -dimensional space. On the left-hand side, we integrate over the same set of planes (up to a null set) stepwise, we step from one j-dimensional subspace in the ambient $(j+1)$ -dimensional space to the next, and in each such subspace, we consider all $(j-1)$ -dimensional subspaces. The above identity holds since we are using probability measures on each nested Grassmannian. Applying the latter iteratively, we get

$$ \begin{align*} & \int_{F^{n}} \widetilde{G}( \eta) d \eta \\ &= \int_{G_{n,n-1} } \int_{G_{ F_{n-1}, n-2 }} \ldots \int_{G_{ F_2, 1 }} \widetilde{G} (F_{1}, \ldots , F_{n-1}) d F_1\ldots d F_{n-2} d F_{n-1} \\ &= \int_{G_{n,n-1} } \int_{G_{ F_{n-1}, n-2 }} \ldots \int_{G_{ F_2, 1}} G (F_{i_1}, \ldots, F_{i_r}) dF_1\ldots d F_{n-2} d F_{n-1} \\ & = \int_{G_{n,n-1} } \ldots \int_{G_{ F_{i_1+1}, i_1 }} G (F_{i_1}, \ldots, F_{i_r}) \left( \int_{G_{F_{i_1},i_1-1}} \ldots \int_{G_{F_2,1}} dF_1 \ldots dF_{i_1-1} \right) d F_{i_1} \ldots dF_{n-1} \\ &= \int_{G_{n,n-1} } \ldots \int_{G_{ F_{i_1+1}, i_1 }} G (F_{i_1}, \ldots, F_{i_r}) d F_{i_1} \ldots dF_{n-1} \\ &= \int_{G_{n,n-1} } \ldots \left( \int_{G_{ F_{i_2}, i_2-1 }} \ldots \int_{G_{ F_{i_1+1}, i_1}} G(F_{i_1}, \ldots, F_{i_r}) d F_{i_1} \ldots dF_{i_2-1} \right) \ldots d F_{n-1} \\ &= \int_{G_{n,n-1} } \ldots \left( \int_{G_{F_{i_2}, i_1 }} G (F_{i_1}, \ldots, F_{i_r}) d F_{i_1} \right) \ldots dF_{n-1} \\ &= \cdots \\ &= \int_{G_{n,i_r} } \int_{G_{F_{i_r}, i_{r-1} }} \ldots \int_{G_{ F_{i_2}, i_1 }} G (F_{i_1}, \ldots, F_{i_r}) d F_{i_1}\ldots dF_{i_{r-1}}d F_{i_r} \\ &=\int_{F_{\mathbf{r}}^{n}} G(\xi_{\mathbf{r}}) d \xi_{\mathbf{r}}.\\[-37pt] \end{align*} $$

We now turn to the invariance properties of the functionals $\Phi _{\mathbf {r}}$ and $\Psi _{\mathbf {r}}$ . Although self-contained proofs are possible, they require somewhat involved machinery. Since all of the ingredients are available in the literature [Reference Dann, Paouris and Pivovarov12, Reference Furstenberg and Tzkoni16, Reference Grinberg20], we have chosen to gather the essentials without proofs. For readers less familiar with the relevant work, we will explain the main points behind the affine invariance of the functionals $\Phi _{{[k]}}(K)$ and $\Psi _{{[k]}}(K)$ along the way. There are two important changes of variables: a “global” change of variables on the Grassmannian $G_{n,k}$ or the flag manifold $F^n_{\mathbf {r}}$ and a “local” change of variables on each element $F\in G_{n,k}$ or $\xi _{\mathbf {r}} \in F^n_{\mathbf {r}}$ .

Let $g\in SL_n$ , $F\in G_{n,k}$ , and $A\subset F$ be a full-dimensional Borel set, then $|g A| = |\det (g|_F)| |A|$ . This determinant of the transformation g restricted to the subspace F, $\det (g|_F)$ , is the Jacobian in the following change of variables:

(20) $$ \begin{align} \int_{gF} f(g^{-1}t) dt = \int_F f(t) |\det(g|_F)| dt. \end{align} $$

Denote it as in [Reference Furstenberg and Tzkoni16] by $\sigma _k(g,F):= |\det (g|_F)| = \frac {|g A|}{|A|}$ .

For the relevant manifolds M considered in this paper, denote by $\sigma _M(g,F)$ the Jacobian determinant in the following change of variables:

(21) $$ \begin{align} \int_M f(F) dF = \int_M f(gF) \sigma_M(g,F) dF. \end{align} $$

Furstenberg and Tzkoni proved in [Reference Furstenberg and Tzkoni16] that

(22) $$ \begin{align} \sigma_{G_{n,k}}( g, F) = \sigma_{k}^{-n} (g,F) \end{align} $$

and

(23) $$ \begin{align} \sigma_{F_{\mathbf{r}}^{n} }(g, \xi_{\mathbf{r}}) = \sigma_{i_{1}}^{-i_{2}}(g, F_1) \sigma_{i_{2}}^{i_{1}-i_{3}}(g, F_2) \cdots \sigma_{i_{r}}^{i_{r-1}-n}(g, F_r) , \end{align} $$

where $\mathbf {r}:= (i_{1}, \ldots , i_{r})$ . The linear invariance of the dual affine quermassintegrals $ \Psi _{[k]} $ now follows immediately. Indeed, for $g\in SL_n$ ,

$$ \begin{align*} \Psi^{kn}_{[k]}(gL) &=\int_{G_{n,k}} |gL\cap F|^n dF = \int_{G_{n,k}} |gL\cap gF|^n \sigma_{G_{n,k}}(g,F) dF \\ &= \int_{G_{n,k}} \left(\sigma_{k}(g,F) |L\cap F|\right)^n \sigma_{k}^{-n} (g,F) dF = \Psi^{kn}_{[k]}(L), \end{align*} $$

where we have used (21), (20) with $f=1_L$ , and (22). Now we turn toward the proof of Theorem 1.1. We start with the case of dual $\mathbf {r}$ -flag affine quermassintegrals.

Proposition 2.3 Let $ 1\leq r \leq n-1$ and $ \mathbf {r} := ( i_{1}, \ldots , i_{r})$ be an increasing sequence of integers between $1$ and $n-1$ . For every compact body L in ${\mathbb R}^{n}$ and every $g \in SL_{n}$ ,

(24) $$ \begin{align} \Psi_{\mathbf{r}} (g L) = \Psi_{\mathbf{r}} (L). \end{align} $$

Proof Let us start by expressing $\sigma _{F_{\mathbf {r}}^{n} }(g, \xi _{\mathbf {r}})$ in terms of sections. For this, note that

$$ \begin{align*}\sigma_{i_{j}}(g, F_{j}) = \frac{ |g( L \cap F_{j})|}{ | L\cap F_{j}|},\end{align*} $$

where as a subset of $F_{j}$ we use the section $ L \cap F_{j}$ . By (23) with $i_0=0$ and $i_{r+1}=n$ , we have

$$ \begin{align*}\sigma_{F_{\mathbf{r}}^{n} } ( g, \xi_{\mathbf{r}}):= \prod_{j=1}^{r} \sigma_{i_{j}}^{-i_{j+1}+ i_{j-1}} (g, F_{j})=\prod_{j=1}^{r} \frac{ |L \cap F_{j}|^{ i_{j+1}- i_{j-1}} }{ |gL \cap g F_{j}|^{ i_{j+1}-i_{j-1}}} .\end{align*} $$

Using the change of variables (21) with the above expression for $\sigma _{F_{\mathbf {r}}^{n} }$ , yields

$$ \begin{align*} \Psi_{\mathbf{r}}^{ni_{r}} ( gL) &= \int_{F_{\mathbf{r}}^{n}} \prod_{j=1}^{r} | gL \cap F_{j}|^{ i_{j+1}- i_{j-1}} d\xi_{\mathbf{r}} \\ &= \int_{F_{\mathbf{r}}^{n}} \prod_{j=1}^{r} | gL \cap g F_{j}|^{ i_{j+1}- i_{j-1}} \sigma_{F_{\mathbf{r}}^{n}} (g,\xi_{\mathbf{r}}) d\xi_{\mathbf{r}} \\ &= \int_{F_{\mathbf{r}}^{n}} \prod_{j=1}^{r} | gL \cap g F_{j}|^{ i_{j+1}- i_{j-1}}\prod_{j=1}^{r} \frac{ | L \cap F_{j}|^{ i_{j+1}- i_{j-1}} }{ |gL \cap g F_{j}|^{i_{j+1}- i_{j-1}}} d\xi_{\mathbf{r}} \\ &=\int_{F_{\mathbf{r}}^{n}} \prod_{j=1}^{r} | L \cap F_{j}|^{ i_{j+1}- i_{j-1}} d\xi_{\mathbf{r}} \\ &=\Psi_{\mathbf{r}}^{ni_{r}} (L).\\[-34pt] \end{align*} $$

To recall the linear invariance of the operator $\Phi _{[k]}$ , we follow Grinberg [Reference Grinberg20]. Observe that for $F\in G_{n,k}$ and $g\in SL_n$ upper-triangular with respect to the decomposition $\mathbb {R}^n=F+F^{\perp }$ , we have

$$ \begin{align*}|P_F ( g^t L )| = |g P_F L| = |\det(g|_F)| |P_F L| = \sigma_k(g,F) |P_F L|,\end{align*} $$

where $g^t$ stands for the transpose of g. While for $l\in SO_n$ , we have $P_F(l^t L)=P_{l F}(L)$ . Since any $g\in SL_n$ can be written as a product of a rotation and an upper-triangular matrix, combining the two observations yields the following.

Lemma 2.4 [Reference Grinberg20]

Let L be a compact body in ${\mathbb R}^{n}$ , $F\in G_{n,k}$ , and $g \in SL_{n}$ . Then

(25) $$ \begin{align} | P_{F} (g^t L)| = |P_{gF} L| \sigma_{k}(g,F). \end{align} $$

The linear invariance of the affine quermassintegrals $ \Phi _{[k]} $ can now be seen as follows: let $g\in SL_n$ ,

$$ \begin{align*}\Phi^{-kn}_{[k]}(g^t L)=\int_{G_{n,k}} | P_{F} (g^t L)|^{-n} dF = \int_{G_{n,k}} |P_{gF} L|^{-n} \sigma_{k}^{-n} (g,F) dF = \Phi^{-kn}_{[k]} (L),\end{align*} $$

where we have used (25) and (21) taking into account (22).

Proposition 2.5 Let $ 1\leq r \leq n-1$ and $ \mathbf {r} := ( i_{1}, \ldots , i_{r})$ be an increasing sequence of integers between $1$ and $n-1$ . Let A be an affine volume preserving map in ${\mathbb R}^{n}$ . Then, for every compact body L in ${\mathbb R}^{n}$ ,

(26) $$ \begin{align} \Phi_{\mathbf{r}} (AL) = \Phi_{\mathbf{r}} (L). \end{align} $$

Proof We will first prove the theorem in the case $A:= g\in SL_{n}$ . Using (25) for the projection onto each $F_j$ , (23), and making the change of variables (21), we get

$$ \begin{align*} \Phi^{-n i_r}_{\mathbf{r}} ( g^t L) &= \int_{F_{\mathbf{r}}^{n}} \prod_{j=1}^{r} | P_{F_{j}}(g^t L)|^{- i_{j+1}+ i_{j-1}} d\xi_{\mathbf{r}} \\ &= \int_{F_{\mathbf{r}}^{n}} \prod_{j=1}^{r} \left( |P_{g F_j} L| \sigma_{i_j}(g,F_j) \right)^{- i_{j+1}+ i_{j-1}} d\xi_{\mathbf{r}} \\ &= \int_{F_{\mathbf{r}}^{n}} \prod_{j=1}^{r} |P_{g F_j} L|^{- i_{j+1}+ i_{j-1}} \prod_{j=1}^{r} \sigma^{- i_{j+1}+ i_{j-1}}_{i_j}(g,F_j) d\xi_{\mathbf{r}} \\ &= \int_{F_{\mathbf{r}}^{n}} \prod_{j=1}^{r} |P_{g F_j} L|^{- i_{j+1}+ i_{j-1}} \sigma_{F_{\mathbf{r}}^{n} } ( g, \xi_{\mathbf{r}}) d\xi_{\mathbf{r}} \\ &= \int_{F_{\mathbf{r}}^{n}} \prod_{j=1}^{r} | P_{F_{j}} L|^{- i_{j+1}+i_{j-1}} d\xi_{\mathbf{r}} \\ &= \Phi^{-n i_r}_{\mathbf{r}} (L). \end{align*} $$

The general case follows easily. This proves (26).

The proof of Theorem 1.1 is now complete.

3 Inequalities

We start by proving an extension of the inequality of Busemann–Straus and Grinberg (8) to flag manifolds.

Proposition 3.1 Let $ 1 < r \leq n-1$ and $ \mathbf {r} := (i_{1}, \ldots , i_{r})$ be an increasing sequence of integers between $1$ and $n-1$ . Then, for every compact body L in $\mathbb {R}^n$ ,

(27) $$ \begin{align} \Psi_{\mathbf{r}} (L) \leq \Psi_{\mathbf{r}} (r_{L } B_{2}^{n}) \end{align} $$

with equality if and only if L is an origin-symmetric ellipsoid (up to a set of measure zero).

Proof Inequality (8) implies that for every n, every $ E\in G_{n,m}$ , any $1\leq \ell \leq m-1$ and every compact body $L \subseteq {\mathbb R}^{n}$ ,

(28) $$ \begin{align} \int_{G_{E,\ell} } | (L \cap E) \cap F |^{ m} d \mu_{G_{E, \ell}} (F) \leq \frac{\omega_{\ell}^m}{\omega_m^{\ell}} | L \cap E|^{\ell} , \end{align} $$

with equality iff $L \cap E $ is an ellipsoid (up to a measure $0$ set; see [Reference Gardner18]). Using (18) and (28), we have that

$$ \begin{align*} \Psi_{\mathbf{r}}^{ i_{r} n} (L) & = \int_{F_{\mathbf{r}}^{n}} \prod_{j=1}^{r} |L\cap F_{{j}} |^{ i_{j+1}- i_{j-1}} d \xi_{\mathbf{r}} \\ & = \int_{ G_{n, i_{r}} } \int_{G_{F_{r}, i_{r-1}}} \ldots \int_{ G_{F_{2}, i_{1}}} \prod_{j=1}^{r} |L\cap F_{{j}} |^{ i_{j+1}- i_{j-1}} dF_{1} \ldots dF_{r-1} dF_{r} \\ & = \int_{ G_{n, i_{r}} } \int_{G_{F_{r}, i_{r-1}}} \ldots\int_{G_{F_{3}, i_{2}}} \prod_{j=2}^{r} |L\cap F_{{j}} |^{ i_{j+1}- i_{j-1}} \times \\ & \qquad \times \left( \int_{ G_{F_{2}, i_{1}}} |L\cap F_{1} |^{ i_{2}} dF_{1} \right) dF_{2} \ldots dF_{r-1} dF_{r}\\ & = \int_{ G_{n, i_{r}} } \int_{G_{F_{r}, i_{r-1}}} \ldots\int_{G_{F_{3}, i_{2}}} \prod_{j=2}^{r} |L\cap F_{{j}} |^{i_{j+1}- i_{j-1}} \times \\ & \qquad \times \left( \int_{ G_{F_{2}, i_{1}}} |(L\cap F_{2}) \cap F_{1} |^{ i_{2}} dF_{1} \right) dF_{2} \ldots dF_{r-1} dF_{r} \\ & \leq \frac{\omega_{i_1}^{i_2}}{\omega_{i_2}^{i_1}} \int_{ G_{n, i_{r}} } \int_{G_{F_{r}, i_{r-1}}} \ldots\int_{G_{F_{3}, i_{2}}} \prod_{j=2}^{r} |L\cap F_{{j}} |^{ i_{j+1}- i_{j-1}} \times \\ &\qquad \times |L \cap F_{2} |^{ i_{1}} dF_{2} \ldots dF_{r-1} dF_{r} \\ &\leq \cdots \\ &\leq |L|^{i_{r}} \prod_{j=1}^{r} \frac{\omega_{i_j}^{i_{j+1}}}{\omega_{i_{j+1}}^{i_j}}. \end{align*} $$

The last inequality is an equality only when L is an origin-symmetric ellipsoid, up to a set of measure zero (see [Reference Gardner18]). Since for the Euclidean ball all inequalities in the previous chain are actually equalities, we can compute the constants and by the linear-invariance property established by Furstenberg and Tzkoni, we conclude the proof.

Our next result is a type of Blaschke–Santaló and reverse Blaschke–Santaló inequality for $\mathbf {r}$ -flag affine quermassintegrals. These inequalities concern the volume of the polar body. For a compact set L, we define the polar body $L^{\circ }$ (with respect to the origin) as the convex body

$$ \begin{align*}L^{\circ} := \{ x\in {\mathbb R}^{n} : \langle x, y\rangle\leq 1 , \ \forall y \in L \}.\end{align*} $$

It is straightforward to check the following inclusion: for every compact set L in ${\mathbb R}^{n}$ and $F\in G_{n,k}$ ,

(29) $$ \begin{align} P_{F} L^{\circ} \subseteq \left(L\cap F\right)^{\circ}. \end{align} $$

If, in addition, L is convex and $0$ is in the interior of L,

(30) $$ \begin{align} P_{F} L^{\circ} = \left( L\cap F\right)^{\circ}. \end{align} $$

Recall that the Blaschke–Santaló inequality (e.g., [Reference Gardner17, Reference Schneider49]) states that for every origin-symmetric convex body L in $\mathbb R^{n}$ ,

(31) $$ \begin{align} |L| | L^{\circ} | \leq |B_{2}^{n}|^{2}. \end{align} $$

Moreover, (31) holds when L is a convex body and $L^{\circ }$ is centered, i.e., the centroid of $L^{\circ }$ is at the origin (see [Reference Schneider49]). We do not know the first reference for (31) in case of compact sets; (31) for centered star-shaped sets appears in [Reference Lutwak33]. For origin-symmetric compact sets with interior points, (31) follows from a more general result in [Reference Cordero-Erausquin, Fradelizi, Paouris and Pivovarov10] (as a limiting case of Theorem 1.1 with $f=1_{L^{\prime }}$ , where $L^{\prime }$ is the closure of the interior of L). For a general proof, see [Reference Milman and Yehudayoff38]. An approximate reverse form of this inequality is known as the Bourgain–Milman theorem [Reference Bourgain and Milman5]: for every compact, convex set L with $ 0 \in \mathrm {int}(L)$ ,

(32) $$ \begin{align} |L| |L^{\circ} | \geq c^{n}|B_{2}^{n}|^{2}, \end{align} $$

for some absolute constant $c>0$ . For further background and alternate proofs of this inequality, see [Reference Giannopoulos, Paouris and Vritsiou19, Reference Kuperberg26, Reference Milman40, Reference Nazarov41].

The next proposition is the aforementioned Blaschke–Santaló inequality and its (approximate) reversal in the setting of $\mathbf {r}$ -flag manifolds:

Proposition 3.2 Let $ 1 \leq r \leq n-1$ and $ \mathbf {r} := ( i_{1}, \ldots , i_{r})$ be an increasing sequence of integers between $1$ and $n-1$ . Then, for an origin-symmetric compact body L in ${\mathbb R}^{n}$ ,

(33) $$ \begin{align} \Phi_{\mathbf{r}} (L^{\circ}) \Psi_{\mathbf{r}} (L) \leq \Phi_{\mathbf{r}} ( B_{2}^{n} ) \Psi_{\mathbf{r}} (B_{2}^{n}). \end{align} $$

Moreover, if L is a convex body in ${\mathbb R}^{n}$ with $ 0 \in \mathrm { int}(L)$ , we have that

(34) $$ \begin{align} \Phi_{\mathbf{r}} (L^{\circ}) \Psi_{\mathbf{r}} (L) \geq c \, \Phi_{\mathbf{r}} ( B_{2}^{n} ) \Psi_{\mathbf{r}} (B_{2}^{n}) , \end{align} $$

where $c>0$ is an absolute constant (from the reverse Santaló inequality (32)).

Proof First, note that

$$ \begin{align*}\Phi_{\mathbf{r}}^{i_{r} n} ( B_{2}^{n} ) \Psi_{\mathbf{r}}^{i_{r} n} (B_{2}^{n}) = \left( \prod_{j=1}^{r} | B_{2}^{n} \cap F_j|^{i_{j+1}- i_{j-1}} \right)^{2},\end{align*} $$

where $F_j$ are $i_j$ -dimensional subspaces as defined in Section 1.2 and the expression is independent of $ \xi _{\mathbf {r}} := (F_{1}, \ldots , F_{r})$ . Replacing $L \cap F_j$ by its convex hull in $F_j$ , using the Blaschke–Santaló inequality (31) and (29), we have

$$ \begin{align*} \Psi_{\mathbf{r}}^{i_{r}n} (L)&=\int_{F_{\mathbf{r}}^{n}}\prod_{j=1}^{r}|L \cap F_j|^{ i_{j+1}- i_{j-1}} d \xi_{\mathbf{r}} \\ &\leq \int_{F_{\mathbf{r}}^{n}}\left(\prod_{j=1}^{r} | B_{2}^{n} \cap F_j|^{i_{j+1}- i_{j-1}}\right)^{2} \prod_{j=1}^{r} \frac{1}{ |(L \cap F_j)^{\circ}|^{ i_{j+1}- i_{j-1}} } d \xi_{\mathbf{r}} \\ & \leq \int_{F_{y r}^{n}}\left( \prod_{j=1}^{r} | B_{2}^{n} \cap F_j|^{i_{j+1}-i_{j-1}}\right)^{2} \prod_{j=1}^{r} \frac{1}{|P_{F_j}L^{\circ}|^{ i_{j+1}- i_{j-1}} } d \xi_{\mathbf{r}} \\ & = \Phi_{\mathbf{r}}^{i_{r} n} ( B_{2}^{n} ) \Psi_{\mathbf{r}}^{i_{r} n} (B_{2}^{n}) \Phi_{\mathbf{r}}^{-i_{r}n} (L^{\circ}). \end{align*} $$

On the other hand, using the reverse Blaschke–Santaló inequality (32), (30), and (9), we get

$$ \begin{align*} \Psi_{\mathbf{r}}^{i_{r}n}(L) & = \int_{F_{\mathbf{r}}^{n}}\prod_{j=1}^{r} |L \cap F_j|^{ i_{j+1}- i_{j-1}} d \xi_{\mathbf{r}}\\ &\geq c^{ \sum_{j=1}^{r} i_{j} (i_{j+1}-i_{j-1}) } \int_{F_{\mathbf{r}}^n } \left( \prod_{j=1}^{r} | B_{2}^{n} \cap F_j|^{i_{j+1}- i_{j-1}} \right)^{2}\prod_{j=1}^{r} \frac{1}{|(L \cap F_j)^{\circ}|^{ i_{j+1}- i_{j-1}} } d \xi_{\mathbf{r}} \\ & = c^{i_{r}n} \int_{F_{\mathbf{r}}^{n}}\left( \prod_{j=1}^{r} | B_{2}^{n} \cap F_j|^{i_{j+1}- i_{j-1}}\right)^{2} \prod_{j=1}^{r} \frac{1}{ | P_{F_j}L^{\circ}|^{ i_{j+1}-i_{j-1}} } d \xi_{\mathbf{r}}\\ &= c^{i_{r}n} \Phi_{\mathbf{r}}^{i_{r} n} ( B_{2}^{n} ) \Psi_{\mathbf{r}}^{i_{r} n} (B_{2}^{n})\Phi_{\mathbf{r}}^{-i_{r}n}(L^{\circ}). \end{align*} $$

The proof is complete.

We now turn to the proof of (5) in the setting of r-flag manifolds. Let us first prove an analogue of (5) for r-flag affine quermassintegrals at the expense of a universal constant using known techniques. The case $r=1$ of the following corollary has been proved in [Reference Paouris and Pivovarov42] before.

Corollary 3.3 Let $ 1 \leq r \leq n-1$ and $ \mathbf {r} := ( i_{1}, \ldots , i_{r})$ be an increasing sequence of integers between $1$ and $n-1$ . Then, for every convex body L,

$$ \begin{align*} \Phi_{\mathbf{r}} (L) \geq c \, \Phi_{\mathbf{r}} (r_{L} B_{2}^{n}) , \end{align*} $$

where $c>0$ is an absolute constant.

Proof As $\Phi _{\mathbf {r}}(L)$ is translation-invariant, we may assume that L is centered. The Blaschke–Santaló inequality implies $r_L r_{L^{\circ }}\leq 1$ , so with (34) and (27), we obtain

$$ \begin{align*}\Phi_{\mathbf{r}} (L) \geq c \frac{ \Phi_{\mathbf{r} } (B_{2}^{n}) \Psi_{\mathbf{r}} (B_{2}^{n}) }{ \Psi_{\mathbf{r}}(L^{\circ})} \geq c \frac{ \Phi_{\mathbf{r} } (B_{2}^{n}) \Psi_{\mathbf{r}} (B_{2}^{n}) }{ \Psi_{\mathbf{r}}( r_{L^{\circ}} B_{2}^{n})} = \frac{c}{ r_{L^{\circ}}} \Phi_{\mathbf{r}} ( B_{2}^{n}) \geq c \, \Phi_{\mathbf{r}} (r_{L} B_{2}^{n}).\end{align*} $$

The proof is complete.

The constant in the above corollary can be made $1$ using (5) and an argument similar to that of Proposition 3.1, as we argue next.

Proposition 3.4 Let $ 1 < r \leq n-1$ and $ \mathbf {r} := (i_{1}, \ldots , i_{r})$ be an increasing sequence of integers between $1$ and $n-1$ . Then, for every convex body L in $\mathbb {R}^n$ ,

(35) $$ \begin{align} \Phi_{\mathbf{r}} (L) \geq \Phi_{\mathbf{r}} (r_{L } B_{2}^{n}) \, , \end{align} $$

with equality if and only if L is an ellipsoid.

Proof First, note that (5) can be written equivalently as

(36) $$ \begin{align} \int_{G_{n,k}}\left\lvert P_E L\right\rvert^{-n}dE \leq \omega_n^k \omega_k^{-n} |L|^{-k} \, , \end{align} $$

and

(37) $$ \begin{align} \Phi_{\mathbf{r}}^{- i_{r} n} (r_L B^n_2) = \prod_{j=1}^{r} |r_L B^{i_j}_2|^{{i_{j-1} -i_{j+1}}} = \left( \prod_{j=1}^{r} \omega_{i_j}^{i_{j-1} -i_{j+1}} \right) \omega_n^{i_r} |L|^{-i_r} \,. \end{align} $$

We repeatedly use the following simple reformulation of (36) that for every n, every $ E\in G_{n,m}$ , and any $1\leq \ell \leq m-1$ ,

(38) $$ \begin{align} \int_{G_{E,\ell} } | P_F L |^{ -m} dF = \int_{G_{E,\ell} } | P_F (P_E L ) |^{ -m} dF \leq \omega_m^{\ell} \omega_\ell^{-m} \, |P_E L|^{-\ell}. \end{align} $$

Using (18) and (36)–(38), we obtain

$$ \begin{align*} &\Phi_{\mathbf{r}}^{- i_{r} n} (L)\\ &\quad = \int_{F_{\mathbf{r}}^{n}} \prod_{j=1}^{r} |P_{F_{j}} L|^{i_{j-1}-i_{j+1}} d \xi_{\mathbf{r}} \\ &\quad = \int_{ G_{n, i_{r}} } \ldots\int_{G_{F_{3}, i_{2}}} \prod_{j=2}^{r} |P_{F_j}L|^{i_{j-1}-i_{j+1}} \left( \int_{ G_{F_{2}, i_{1}}} |P_{F_1}(P_{F_2}L)|^{-i_{2}} dF_{1} \right) dF_{2} \ldots dF_{r} \\ &\quad \leq \omega^{i_1}_{i_2} \omega_{i_1}^{-i_2} \int_{ G_{n, i_{r}} } \ldots\int_{G_{F_{3}, i_{2}}} \prod_{j=2}^{r} |P_{F_j}L|^{i_{j-1}-i_{j+1}} \left( |P_{F_2}L|^{-i_{1}} \right) dF_{2} \ldots dF_{r} \\ &\quad = \omega^{i_1}_{i_2} \omega_{i_1}^{-i_2} \int_{ G_{n, i_{r}} } \ldots\int_{G_{F_{4}, i_{3}}} \prod_{j=3}^{r} |P_{F_j}L|^{i_{j-1}-i_{j+1}} \left( \int_{G_{F_{3}, i_{2}}} |P_{F_2}(P_{F_3}L)|^{-i_3} dF_{2} \right) dF_{3} \ldots dF_{r} \\ &\quad \leq (\omega^{i_1}_{i_2} \omega_{i_1}^{-i_2}) (\omega^{i_2}_{i_3} \omega_{i_2}^{-i_3}) \int_{ G_{n, i_{r}} } \ldots\int_{G_{F_{4}, i_{3}}} \prod_{j=3}^{r} |P_{F_j}L|^{i_{j-1}-i_{j+1}} \left( |P_{F_3}L|^{-i_2} \right) dF_{3} \ldots dF_{r} \\ &\quad \cdots\\ &\quad\leq \prod_{j=1}^{r} \omega^{i_j}_{i_{j+1}} \omega_{i_j}^{-i_{j+1}} \quad |L|^{-i_{r}} = \left( \prod_{j=1}^{r} \omega_{i_j}^{i_{j-1} -i_{j+1}} \right) \omega_n^{i_r} \quad |L|^{-i_{r}} = \Phi_{\mathbf{r}}^{- i_{r} n} (r_L B^n_2) \, , \end{align*} $$

where for the penultimate equality we have used the convention that $i_0=0$ and $i_{r+1}=n$ . This proves (35). For the Euclidean ball, all inequalities in the above argument are equalities [Reference Milman and Yehudayoff38]. By the affine invariance of $\Phi _{\mathbf {r}}$ , the equality holds only when L is an ellipsoid.

The next proposition shows that for a convex body L in $\mathbb {R}^n$ , all the quantities $\Phi _{\mathbf {r}}(L)$ lie between the volume-radius $r_{L}$ and the mean width $W(L)=2\int _{S^{n-1}}h_L(\theta )d\sigma (\theta )$ ; here, $h_L$ is the support function of L, i.e., $h_L(\theta )=\sup _{x\in L}\langle x,\theta \rangle $ . We also set

(39) $$ \begin{align} W_{L} := \inf_{T\in SL_{n}} W(T L). \end{align} $$

It will be convenient to use Urysohn’s inequality ( $k=1$ in (4)) in the following form. For any convex body K in $\mathbb {R}^n$ ,

$$ \begin{align*} \left(\frac{\left\lvert K\right\rvert}{\left\lvert B_2^n\right\rvert}\right)^{1/n}\leq \frac{W(K)}{2}. \end{align*} $$

Proposition 3.5 Let $ 1 \leq r \leq n-1$ and $ \mathbf {r} := ( i_{1}, \ldots , i_{r})$ be an increasing sequence of integers between $1$ and $n-1$ . Then, for any convex body L in ${\mathbb R}^{n}$ ,

(40) $$ \begin{align} r_L \leq \frac{ \Phi_{\mathbf{r}}(L) }{ \Phi_{\mathbf{r}} (B_{2}^{n}) }\leq \frac{W_{L}}{2}. \end{align} $$

Proof The left-most inequality follows from (35). Next, for $s>0$ , Hölder’s inequality and the fact that $h_{L} (x) = h_{P_{F}L}(x)$ for $x\in F$ , give

(41) $$ \begin{align} \left(\int_{G_{n,k}}W(P_{F}L)^{-s}dF\right)^{-1/s}\leq \int_{G_{n,k}} W(P_{F} L) \, d F = W(L). \end{align} $$

Toward proving the right-hand side, we write

(42) $$ \begin{align} \Phi_{\mathbf{r}}^{-ni_r}(L) & = \int_{ G_{n, i_{r}} } \ldots\int_{G_{F_{3}, i_{2}}}\int_{G_{F_{2}, i_{1}}} \prod_{j=1}^{r} |P_{F_j}L|^{i_{j-1}-i_{j+1}} dF_{1}dF_{2} \ldots dF_{r}. \end{align} $$

Working with the inner-most integral (recall $i_0=0$ ), we invoke Urysohn’s inequality, followed by (41) to obtain

$$ \begin{align*} \int_{G_{F_2,i_1}}\frac{1}{|P_{F_1}L|^{i_2}}dF_{1} \geq \int_{G_{F_2,i_1}} \frac{2^{i_1i_2}}{\omega_{i_1}^{i_2}W(P_{F_1}L)^{i_1i_2}}dF_{1} \geq \frac{2^{i_1i_2}}{\omega_{i_1}^{i_2}W(P_{F_2}L)^{i_1i_2}}. \end{align*} $$

Similarly, by Urysohn’s inequality,

$$ \begin{align*} \frac{1}{|P_{F_2}L|^{i_3-i_1}}\geq \frac{2^{i_2(i_3-i_1)}}{\omega_{i_2}^{i_3-i_1}W(P_{F_2}L)^{i_2(i_3-i_1)}}. \end{align*} $$

Inserting the last two inequalities into (42) and applying the same argument iteratively, we have

$$ \begin{align*} \Phi_{\mathbf{r}}^{-ni_r}(L) &\geq \int_{ G_{n, i_{r}} } \ldots\int_{G_{F_{4}, i_{3}}} \prod_{j=3}^{r} |P_{F_j}L|^{i_{j-1}-i_{j+1}} \int_{G_{F_3,i_2}}\frac{2^{i_2i_3}}{\omega_{i_2}^{i_3-i_1}\omega_{i_1}^{i_2}W(P_{F_2}L)^{i_2i_3}} dF_{2}dF_{3} \ldots dF_{r}\\ &\geq \int_{ G_{n, i_{r}} } \ldots\int_{G_{F_{4}, i_{3}}} \prod_{j=3}^{r} |P_{F_j}L|^{i_{j-1}-i_{j+1}} \frac{2^{i_2i_3}}{\omega_{i_2}^{i_3-i_1}\omega_{i_1}^{i_2}W(P_{F_3}L)^{i_2i_3}} dF_{3} \ldots dF_{r}\\ &\geq \ldots\\ &\geq 2^{ni_r}\Phi^{-ni_r}_{\mathbf{ r}}(B_2^n)\int_{G_{n,i_r}}\frac{1}{W(P_{F_r}L)^{ni_r}}dF_r\\ &\geq 2^{ni_r}\Phi^{-ni_r}_{\mathbf{ r}}(B_2^n)W(L)^{-ni_r}.\\ \end{align*} $$

In the above argument, we may replace L by $TL$ , where $T\in SL_n$ . Since the left-hand side of this inequality remains the same for all T by Theorem 1.1, we may take the infimum over all T on the right-hand side. This completes the proof.

We conclude this subsection with a discussion of inequalities of isomorphic nature. For convex bodies L in $\mathbb {R}^n$ , we define the Banach–Mazur distance to the Euclidean ball $B_2^n$ by

$$ \begin{align*}d_{BM}(L) := \inf\left\{ ab : a>0,b>0, \frac{1}{b} B_{2}^{n} \subseteq T(L-L) \subseteq a B_{2}^{n}, T\in GL_{n} \right\}.\end{align*} $$

For origin-symmetric convex bodies, this coincides with the standard notion of Banach–Mazur distance (for more information, see, e.g., [Reference Tomczak-Jaegermann52]).

Proposition 3.6 Let $ 1 \leq r \leq n-1$ and $ \mathbf {r} := ( i_{1}, \ldots , i_{r})$ be an increasing sequence of integers between $1$ and $n-1$ . Then, for any convex body L in $\mathbb {R}^n$ ,

(43) $$ \begin{align} \Phi_{\mathbf{r}}(L) \leq c_1 \min\left\{ \sqrt{\frac{n}{i_{r}}}, \log{(1+ d_{BM}(L))} \right\} \Phi_{\mathbf{r}} (r_{L} B_{2}^{n}), \end{align} $$

and, if L is also symmetric, then

(44) $$ \begin{align} \Psi_{\mathbf{r}}(L) \geq \frac{c_2}{\min\left\{ \sqrt{\frac{n}{i_{r}}} , \log{(1+ d_{BM}(L))} \right\} }\Psi_{\mathbf{r}} (r_{L} B_{2}^{n}), \end{align} $$

where $c_1,c_2>0$ are absolute constants.

The proof relies on several different tools. We draw on ideas from Dafnis and the second-named author [Reference Dafnis and Paouris11] to exploit the affine invariance of $\Phi _{\mathbf {r}}(L)$ , $\Psi _{\mathbf { r}}(L)$ by using appropriately chosen affine images of L. To this end, recall the following fundamental theorem, which combines work of Figiel–Tomczak-Jaegermann [Reference Figiel and Tomczak-Jaegermann13], Lewis [Reference Lewis28], Pisier [Reference Pisier45], and Rogers–Shephard [Reference Rogers and Shephard47] (see Theorem 1.11.5 on page 52 in [Reference Brazitikos, Giannopoulos, Valettas and Vritsiou6]).

Theorem 3.7 Let L be a centered convex body in $\mathbb {R}^n$ . Then there exists a linear map $T\in SL_{n}$ such that

$$ \begin{align*} W(T L) \leq c \log\{ 1+ d_{BM} (L)\} \sqrt{n}|L|^{1/n}, \end{align*} $$

for some absolute constant $c> 0$ .

We will also use recent results on isotropic convex bodies. For background, the reader may consult [Reference Brazitikos, Giannopoulos, Valettas and Vritsiou6], but we will recall all facts that we need here. To each convex body $M\subseteq \mathbb {R}^n$ with unit volume, one can associate an ellipsoid $Z_{2}(M)$ , called the $L_{2}$ -centroid body of M, which is defined by its support function as

$$ \begin{align*}h_{Z_{2}(M)} (\theta) := \left( \int_{M} | \langle x, \theta \rangle |^{2} dx \right)^{\frac{1}{2}}.\end{align*} $$

The isotropic constant of M is defined by $L_{M} := r_{Z_{2}(M)}$ . We say that M is isotropic if it is centered and $Z_{2}(M) = L_{M} B_{2}^{n}$ . Fix an isotropic convex body M and a k-dimensional subspace F. Ball [Reference Ball3] proved that

(45) $$ \begin{align} |M \cap F^{\perp} |^{\frac{1}{ k}} \geq \frac{c}{L_{M}}; \end{align} $$

a corresponding inequality for projections,

(46) $$ \begin{align} |P_{F} M| \leq \left(c \frac{n}{k} L_{M}\right)^{k}, \end{align} $$

follows immediately from (45) and the Rogers–Shephard inequality [Reference Rogers and Shephard47]:

$$ \begin{align*}|P_{F} M| |M\cap F^{\perp}| \leq {n\choose k}.\end{align*} $$

Next, we recall a variant of $\Psi _{[k]}(M)$ studied by Dafnis and the second-named author [Reference Dafnis and Paouris11]. For every $ 1\leq k \leq n-1$ and a compact body M in ${\mathbb R}^{n}$ with $|M|=1$ , we define

(47) $$ \begin{align} \widetilde{\Phi}_{[k]}(M) := \left( \int_{G_{n,k}} |M\cap F^{\perp}|^{n} d\mu_{G_{n,k}}(F) \right)^{\frac{1}{nk}}. \end{align} $$

In [Reference Dafnis and Paouris11], it is shown that for every centered convex body M in $\mathbb {R}^n$ of unit volume,

$$ \begin{align*} \frac{c_{1}}{ L_{M}} \leq \widetilde{\Phi}_{[k]}(M) \leq \widetilde{ \Phi}_{[k]}(D_n) \simeq 1 , \end{align*} $$

where $D_n$ is the Euclidean ball of volume one.

We also invoke Klartag’s [Reference Klartag24] fundamental result on perturbations of isotropic convex bodies having a well-bounded isotropic constant.

Theorem 3.8 Let M be a convex body in ${\mathbb R}^{n}$ . For every $\varepsilon \in (0,1)$ , there exists a centered convex body $M_{Kl} \subset {\mathbb R}^{n}$ and a point $x\in {\mathbb R}^{n}$ such that

(48) $$ \begin{align} \frac{1}{ 1+\varepsilon} M_{Kl} \subseteq M + x \subseteq (1+\varepsilon) M_{Kl} \end{align} $$

and

(49) $$ \begin{align} L_{M_{Kl}} \leq \frac{c}{ \sqrt{\varepsilon}}. \end{align} $$

We are now ready to complete the proof.

Proof of Proposition 3.6

By homogeneity of the operators $\Phi _{\mathbf {r}}$ and $\Psi _{\mathbf {r}}$ , we can assume that L has unit volume.

First, we will prove the bound (43) for $\mathbf {r}$ -flag affine quermassintegrals. By translation invariance of projections, we may further assume that L is centered. Bounding $\Phi _{\mathbf {r}}(L)$ by $W(L)$ according to (40), using affine invariance of $\Phi _{\mathbf {r}}$ and reverse Urysohn inequality from Theorem 3.7, we get

(50) $$ \begin{align} \Phi_{\mathbf{r}} (L) \leq c\log( 1+ d_{BM}(L)) \Phi_{\mathbf{r}}(D_n). \end{align} $$

For the Euclidean ball $D_n$ of unit volume, we have $|P_{F} D_{n}|^{\frac {1}{k}}=| D_{n} \cap F|^{\frac {1}{k}}\simeq \sqrt {\frac {n}{k}}$ for every $F\in G_{n,k}$ , so

$$ \begin{align*}\Phi_{\mathbf{r}} ( D_{n}) \simeq \left( \prod_{j=1}^{r} \left(\frac{n}{ i_{j}}\right)^{i_{j} (i_{j+1}-i_{j-1}) }\right)^{\frac{1}{2i_{r} n}}.\end{align*} $$

The AM/GM inequality implies

$$ \begin{align*}\left( \prod_{j=1}^{r} \left( \frac{n}{ i_{j}}\right)^{i_{j} (i_{j+1}-i_{j-1}) }\right)^{\frac{1}{ 2i_{r} n}} \leq \sqrt{\frac{n}{i_{r}n} \sum_{j=1}^{r} \frac{i_{j} (i_{j+1}-j_{j-1})}{i_{j}} }\leq \sqrt{ \frac{n}{ i_{r}} }.\end{align*} $$

Thus,

(51) $$ \begin{align} \Phi_{\mathbf{r}} (D_{n}) = \Psi_{\mathbf{r}}( D_{n}) \simeq \left( \prod_{j=1}^{r} \left( \frac{n}{ i_{j}}\right)^{i_{j} ( i_{j+1}-i_{j-1}) }\right)^{\frac{1}{ 2i_{r} n}} \leq \sqrt{\frac{n}{i_{r}}}. \end{align} $$

Let $K_{1} \subset \mathbb {R}^n$ be a centered convex body and $x\in \mathbb {R}^n$ from the conclusion of Theorem 3.8 corresponding to $\varepsilon =\frac {1}{2}$ . Then (48) implies $1=|L|^{1/n} \geq \frac {2}{3} | K_{1}|^{1/n}$ , while (49) implies $L_{K_1} \simeq 1$ . Let $ K_{2} := \frac {K_{1}}{ |K_{1}|^{\frac {1}{n}} }$ . Then $L_{K_2} = L_{K_1} \simeq 1$ and our choice of $\varepsilon $ and the latter volume bound give

(52) $$ \begin{align} \Phi_{\mathbf{r}}(L)= \Phi_{\mathbf{r}} (L+x) \leq \frac{3}{2} \Phi_{\mathbf{r}}(K_{1}) \leq \frac{9}{4} \Phi_{\mathbf{r}}( K_{2}). \end{align} $$

Affine invariance of $\Phi _{\mathbf {r}}$ (Theorem 1.1) allows us to assume that $K_{2}$ is isotropic. Using (46), (51), $L_{K_2} \simeq 1$ , and (51) one more time, we obtain

$$ \begin{align*} \Phi_{\mathbf{r}}(K_{2}) &= \left( \int_{F_{\mathbf{r}}^{n}} \prod_{j=1}^{r} | P_{F_{j}} K_2|^{-i_{j+1}+i_{j-1}} d\xi_{\mathbf{r}} \right)^{-\frac{1}{i_{r} n}} \\ &\leq \left(\prod_{j=1}^{r} \left( \frac{n}{ i_{j}}\right)^{i_{j}(i_{j+1}-i_{j-1})} \right)^{\frac{1}{i_{r}n}} (cL_{K_{2}})^{\frac{1}{i_{r}n}\sum_{j=1}^{r} i_{j}(i_{j+1}-i_{j-1}) }\\ & \leq c \, L_{K_{2}} \Phi_{\mathbf{r}}(D_n)^{2} \\ &\leq c \, \sqrt{\frac{n}{i_{r}}} \, \Phi_{\mathbf{r}}(D_n). \end{align*} $$

By (52), we have $ \Phi _{\mathbf {r}} (L) \leq c^{\prime } \sqrt {\frac {n}{i_{r}}} \, \Phi _{\mathbf {r}} (D_n) $ , which together with (50) implies (43).

Applying (34), (43) for $L^{\circ }$ , and the Blaschke–Santaló inequality $r_L r_{L^{\circ }}\leq 1$ , we get

$$ \begin{align*} \Psi_{\mathbf{r}} (L) &\geq c\frac{ \Phi_{\mathbf{r}} (B_{2}^{n}) \Psi_{\mathbf{r}}(B_{2}^{n}) }{ \Phi_{\mathbf{r}} ( L^{\circ}) } \\ &\geq \frac{c}{r_{L^{\circ}}} \frac{1}{ \min\left\{ \log{(1+ d_{BM}(L^{\circ}))}, \sqrt{\frac{n}{i_r}}\right\}} \frac{\Phi_{\mathbf{r}} (B_{2}^{n}) \Psi_{\mathbf{r}}(B_{2}^{n}) }{ \Phi_{\mathbf{r}} ( B_{2}^{n})} \\ & \geq \frac{c}{ \min\left\{ \log{(1+ d_{BM}(L))}, \sqrt{\frac{n}{i_{r}}}\right\}} \Psi_{\mathbf{r}} ( r_{L} B_{2}^{n}), \end{align*} $$

where we have also used the identity $d_{BM} (L^{\circ }) = d_{BM}(L)$ for origin-symmetric convex bodies. This proves (44).

4 Flag manifolds and permutations

In this section, we discuss more general quantities involving permutations. We investigate the extent to which $SL_{n}$ -invariance properties established by Furstenberg and Tzkoni [Reference Furstenberg and Tzkoni16] carry over from ellipsoids to compact bodies. In particular, we provide an example of a convex body for which $SL_{n}$ -invariance fails. Nevertheless, we show that for convex bodies, such quantities cannot be too degenerate in the sense that they admit uniform upper and lower bounds, independent of the body. The key ingredient is the notion of M-ellipsoids, introduced by V. D. Milman [Reference Milman40].

The next definition is motivated by the work of Furstenberg and Tzkoni [Reference Furstenberg and Tzkoni16] for ellipsoids.

Definition 4.1 Let $\Pi _{n}$ be the set of permutations of $\{1,2,\ldots ,n\}$ and $\omega \in \Pi _{n}$ . For compact bodies L in $\mathbb {R}^n$ , we define the $\omega $ -flag quermassintegral and dual $\omega $ -flag quermassintegrals as follows: if $\omega (n) \neq n$ , then

(53) $$ \begin{align} \Phi_{\omega} (L) := \left( \int_{F^{n}} \prod_{j=1}^{n-1} | P_{F_{j}} L|^{- \omega(j) + \omega(j+1)-1} d \xi \right)^{-\frac{1}{n( n- \omega(n))}} \end{align} $$

and

(54) $$ \begin{align} \Psi_{\omega} (L) := \left( \int_{F^{n}} \prod_{j=1}^{n-1} | L \cap F_{j}|^{ \omega(j) - \omega(j+1)+1} d \xi \right)^{\frac{1}{ n(n- \omega(n))}}. \end{align} $$

When $\omega (n)=n$ , we set

(55) $$ \begin{align} \Phi_{\omega} (L) := \int_{F^{n}} \prod_{j=1}^{n-1} | P_{F_{j}} L|^{- \omega(j) + \omega(j+1)-1} d \xi \end{align} $$

and

(56) $$ \begin{align} \Psi_{\omega} (L) := \int_{F^{n}} \prod_{j=1}^{n-1} | L \cap F_{j}|^{ \omega(j) - \omega(j+1)+1} d \xi. \end{align} $$

Note that

(57) $$ \begin{align} \sum_{j=1}^{n-1} \left( \omega(j) - \omega(j+1) +1\right) = n-\omega(n)+ \omega(1)-1 \end{align} $$

and

(58) $$ \begin{align} \sum_{j=1}^{n-1} j\left( \omega(j) - \omega(j+1) +1\right) = n(n-\omega(n)). \end{align} $$

Identity (58) guarantees that $\Psi _{\omega }(L)$ and $ \Phi _{\omega }(L)$ are $1$ -homogeneous when $\omega (n)\neq n$ and $0$ -homogeneous when $ \omega (n)=n$ .

The following fact for dual $\omega $ -flag quermassintegrals is from [Reference Furstenberg and Tzkoni16]. For $\omega $ -flag quermassintegrals, it follows, for example, by duality.

Theorem 4.2 Let ${\cal {E}}$ be an ellipsoid in ${\mathbb R}^{n}$ and $\omega \in \Pi _{n}$ . Then

$$ \begin{align*} \Psi_{\omega} ({\cal{E}}) = \Psi_{\omega} (r_{\cal{E}}B_2^n) \ \mathrm{ and} \ \Phi_{\omega} ({\cal{E}}) = \Phi_{\omega} (r_{\cal{E}}B_2^n). \end{align*} $$

An equivalent formulation of the latter result is that for every ellipsoid ${\cal {E}}$ ,

$$ \begin{align*} \Psi_{\omega} ({\cal{E}}) = c_{\omega} |{\cal{E}}|^{\frac{1}{n}}, \ \omega(n) \neq n \ \mathrm{and } \ \Psi_{\omega} ({\cal{E}}) = c_{\omega} , \ \omega(n) = n, \end{align*} $$

where $c_{\omega } $ is a constant that depends only on $\omega $ . An analogous statement holds for $\Phi _{\omega }(\cal {E})$ (see the proof of Proposition 3.2).

The operators $\Psi _{\omega }$ and $\Phi _{\omega }$ are generalizations of $\Psi _{\mathbf {r}}$ and $\Phi _{\mathbf {r}}$ . Indeed, let $1\leq r \leq n-1$ , $1\leq i_{1}< i_{2} < \cdots < i_{r}\leq n-1$ , and $ \mathbf {r}:= (i_{1}, \ldots , i_{r}). $ Define $\omega $ by $\omega (1) = n-i_{1}+1$ , and $\omega (t+1)= \omega (t) +1$ , for $ t\neq i_{j}$ , $1\leq j \leq r$ , and $\omega ( i_{j}+1)= \omega (i_{j}) +1 - i_{j+1}+ i_{j-1}$ for $ 1\leq j \leq r$ . Then $\omega \in \Pi _n$ with $ \omega (i_{j} ) - \omega (i_{j}+1) +1 = i_{j+1}- i_{j-1}$ for $1\leq j\leq r$ and $ \omega ( t) - \omega ( t+1) +1 = 0 , \ t\neq i_{j}$ , for $1\leq j \leq r.$ Since $\omega (n) = n-i_{r}$ , for a compact body L in ${\mathbb R}^{n}$ , we have

$$ \begin{align*} \Psi^{n i_{r}}_{\omega} (L) = \int_{F^{n} } \prod_{j=1}^{n-1} |L \cap F_{j}|^{\omega(j)-\omega(j+1)+1} d \xi = \int_{F^{n} } \prod_{j=1}^{r} |L \cap F_{i_{j}}|^{i_{j+1}-i_{j-1}} d \xi = \Psi^{n i_{r}}_{\mathbf{r}}(L), \end{align*} $$

where, in the last equality, we have used (19). Correspondingly, we also have $ \Phi _{\omega }(L) = \Phi _{\mathbf {r}} (L)$ . In particular, for this permutation $\omega $ , $\Psi _{\omega }(L)$ is $SL_{n}$ -invariant and $\Phi _{\omega }(L)$ is affine invariant. As a particular case of the preceding discussion, let $r=1$ , $i_{1}= k$ , $1\leq k \leq n$ , and let $\omega (1)= n-k+1$ and $\omega (t+1)= \omega (t)+1$ for $t\neq k$ and $ \omega (k+1) = \omega (k) - n +1$ . Then $\Phi _{\omega }(L) = \Phi _{[k]}(L).$

Given that $\Psi _{\mathbf {r}}(L)$ and $\Phi _{\mathbf {r}}(L)$ enjoy invariance properties and arise as permutations, it is natural to investigate the extent to which the invariance from Theorem 4.2 carries over to compact bodies. We do not have a complete answer. However, there are cases outside of those considered above where the invariance holds and also counter-examples where it fails as the next two examples show. It will be convenient to recall that a star-body L is a compact body with $0\in \mathop{\mathrm{int}}(L)$ such that $\alpha x\in L$ whenever $x\in L$ and $\alpha \in [0,1]$ , with continuous radial function $\rho _L(\theta )=\sup \{r\geq 0:r\theta \in L\}$ ( $\theta \in S^{n-1}$ ).

Example 1 Let $n\geq 3$ . Define $\omega $ by $\omega (1)=2$ , $\omega (2)=1$ and $\omega (t)=t$ for all $ 3\leq t \leq n$ . We will show that for every origin-symmetric star-body L in ${\mathbb R}^{n}$ ,

$$ \begin{align*}\Psi_{\omega}(L) = \frac{4}{\pi} .\end{align*} $$

In particular, $\Psi _{\omega }(L)$ is $SL_{n}$ -invariant. Note that our choice of the permutation $\omega $ satisfies

$$ \begin{align*}\omega(1)-\omega(2)+1=2, \quad \omega(2)-\omega(3)+1=-1, \quad \omega(j)-\omega(j+1)+1=0 \text{ for } 3\leq j \leq n-1,\end{align*} $$

or equivalently

$$ \begin{align*}\omega(2)=\omega(1)-1, \quad \omega(3)=\omega(1)+1, \quad \omega(j+1)=\omega(1)+(j-1) \text{ for } 3\leq j \leq n-1.\end{align*} $$

Moreover, $\omega $ is the unique permutation satisfying the latter equations, as $1\leq \omega (j)\leq n$ for all $1\leq j\leq n$ . For a k-dimensional subspace $F_k$ of $\mathbb {R}^n$ , denote by $S_{F_k}$ the unit sphere in $F_k$ . Now, using (19) and recalling the notation for $\rho _L$ above, we compute

$$ \begin{align*} \Psi_{\omega} (L) &= \int_{F^{n}} \prod_{j=1}^{n-1} |L \cap F_{j}|^{\omega(j)-\omega(j+1)+1} d\xi \\ &= \int_{F^{n}} |L \cap F_1|^2 |L \cap F_2|^{-1} d\xi \\ &= \int_{G_{n,2}} |L \cap F_2|^{-1} \int_{G_{F_2,1}} |(L \cap F_2) \cap F_1|^2 dF_1 dF_2 \\ &= \int_{G_{n,2}} |L \cap F_2|^{-1} \int_{S_{F_2}} (2 \rho_{L \cap F_2}(\theta))^2 d\sigma(\theta) dF_2 \\ &= \int_{G_{n,2}} |L \cap F_2|^{-1} \frac{4}{|S_{F_2}|} \int_{S_{F_2}} \rho^2_{L \cap F_2}(\theta) d\theta \, dF_2 \\ &= \frac{4}{\pi} \int_{G_{n,2}} |L \cap F_2|^{-1} |L \cap F_2| dF_2 \\ &= \frac{4}{\pi}. \end{align*} $$

When $n=3$ , for permutations $\omega $ with $\omega (3)=3$ , $\Psi _{\omega }(L)$ are absolute constants. Moreover, the discussion following Theorem 4.2 shows that for three of the remaining four permutations $\omega $ in $\Pi _3$ , $\Psi _{\omega }(L)=\Psi _{\mathbf {r}}(L)$ . Altogether, for $n=3$ , for five out of six permutations $\omega $ , $\Psi _{\omega }(L)$ are $SL_n$ -invariant. The next example shows that for the remaining permutation, the invariance does not carry over for all convex bodies.

Example 2 Let $\omega \in \Pi _3$ with $\omega (1)=1, \, \omega (2)=3$ , and $\omega (3)=2$ . We claim that for a centered cube $Q:=[-1,1]^3$ and the diagonal matrix $D=\mathop{\mathrm{diag}}(1,2,1/2)$ , $\Phi _{\omega }(DQ)>\Phi _{\omega }(Q)$ . Since $D \in SL_3$ , this shows that the operator $\Phi _{\omega }$ is not invariant under volume-preserving transformations.

To show this, we first note that for any convex body $L \subset {\mathbb R}^{3}$

$$ \begin{align*} \Phi^{-3}_{\omega}(L) = \int_{G_{3,2}}\left\lvert P_{F_2}L\right\rvert^{-2}\int_{G_{F_2,1}}\left\lvert P_{F_1} L\right\rvert dF_1dF_2 = \int_{S^{2} } \frac{ W( P_{\phi^{\perp}}L)}{ h_{\Pi L}^{2} (\phi) } d\sigma (\phi), \end{align*} $$

where $h_{\Pi L}(\phi )=\left \lvert P_{\phi ^{\perp }}L\right \rvert $ . For $\theta \in S^{2}$ , we have $h_Q(\theta )=\sum _{i=1}^3 |\theta _i|$ and, for $g\in GL_3$ , $h_{gL}(\theta ) = h_L(g^{t}\theta )$ . We will also use the following facts about projection bodies (see, e.g., [Reference Gardner17]). The projection body of a cube is again a cube, $\Pi Q = 2 Q$ , and for $g\in GL_3$ ,

$$ \begin{align*} \Pi (g L) = | \mathrm{det} g | \, g^{-t} \, \Pi L. \end{align*} $$

Let $ A=[a_{1} \ a_{2} \ a_{3} ] \in SL_3$ with columns $a_i$ . Fix $ \phi \in S^{2}$ . Let $U \in O_3$ be given in column form by $ U = [ u \ v \ \phi ] $ . Since $ U $ is orthogonal, $ U^{t} \phi = e_{3} $ and $ U^{t} \phi ^{\perp } = \mathrm {span}\{ e_{1}, e_{2} \} = \mathbb R^{2} $ . Then

$$ \begin{align*}W( P_{\phi^{\perp}} A Q)= 2 \int_{S_{\phi^{\perp}}} h_{A Q } ( \theta) d\sigma (\theta ) = 2 \int_{S^{1}} h_{AQ} ( U\theta ) d \sigma (\theta ) = 2 \int_{S^{1}} h_{Q} ( A^{t} U\theta ) d \sigma ( \theta ).\end{align*} $$

Thus, denoting by P the orthogonal projection onto ${\mathbb R}^{2}$ , we have

$$ \begin{align*}W( P_{\phi^{\perp}} A Q) = 2 \sum_{i=1}^{3} \int_{S^{1}} | \langle \theta , U^{t} A e_{i} \rangle | d\sigma( \theta ) = 2 \sum_{i=1}^{3} \int_{S^{1}} | \langle \theta , P U^{t} A e_{i} \rangle | d\sigma( \theta ) = \frac{4}{\pi} \sum_{i=1}^{3} \| P U^{t} A e_{i} \|_{2} .\end{align*} $$

One can verify the last equality in this setting by a direct computation as follows. We realize integration over $S^{1}$ as integration over $[0, 2\pi ]$ . Hence, we write $\theta =(\cos \alpha , \sin \alpha )$ , $v \in \mathbb {R}^2$ as $v = \| v \|_2 (\cos \beta , \sin \beta )$ and $\int _{S^{1}} | \langle \theta , v \rangle | d\sigma ( \theta ) = \frac {\| v \|_2}{2\pi } \int _0^{2\pi } | \cos \alpha \cos \beta + \sin \alpha \sin \beta | d\alpha = \frac {\| v \|_2}{2\pi } \int _0^{2\pi } | \cos (\alpha - \beta ) | d\alpha = \frac {\| v \|_2}{2\pi } \int _0^{2\pi } | \cos \alpha | d\alpha = \frac {2}{\pi } \| v \|_2$ . We have that $ A e_{i} = a_{i}$ , $ U^{t} a_{i} = \left ( \langle u, a_{i} \rangle , \langle v, a_{i} \rangle , \langle \phi , a_{i} \rangle \right )^{t} $ and

$$ \begin{align*}\| P U^{t} A e_{i} \|_{2}^{2} = \| U^{t} A e_{i} \|_{2}^{2} - \| (I-P) U^{t} A e_{i} \|_{2}^{2} = \|a_{i}\|_{2}^{2} - \langle \phi, a_{i} \rangle^{2} .\end{align*} $$

Therefore,

$$ \begin{align*} W( P_{\phi^{\perp}} A Q)= \frac{4}{\pi} \sum_{i=1}^{3} \sqrt{ \|a_{i}\|_{2}^{2} - \langle \phi, a_{i} \rangle^{2} }. \end{align*} $$

Moreover, by the aforementioned properties of a centered cube, $ h_{\Pi (A Q) } ( \phi ) = h_{\Pi Q} ( A^{-1} \phi ) = 2 \sum _{i=1}^{3} | \langle A^{-1} \phi , e_{i} \rangle | $ . Thus,

$$ \begin{align*} \Phi^{-3}_{\omega}(AQ) = \frac{1}{\pi} \int_{S^{2}} \frac{ \sum_{i=1}^{3} \sqrt{ \|a_{i}\|_{2}^{2} - \langle \phi, a_{i} \rangle^{2} }}{\left(\sum_{j=1}^{3} | \langle A^{-1} \phi, e_{j} \rangle | \right)^{2} } d\sigma ( \phi). \end{align*} $$

Set $ A:= \mathrm {diag} ( d_{1}, d_{2}, d_{3})$ with $\prod _{i=1}^{3} d_{i}= 1$ and $ d_{i}>0$ . Then the quantity

$$ \begin{align*} {\cal{A}}( d_{1}, d_{2}, d_{3}) := \int_{S^{2}} \frac{\sum_{i=1}^{3} d_{i} \sqrt{ 1- \phi_{i}^{2} }}{ \left(\sum_{j=1}^{3}\frac{|\phi_{j}|}{ d_{j}}\right)^{2} } d\sigma ( \phi) \end{align*} $$

is not constant. Indeed, using MATLAB, for example, one can verify that ${\cal {A}}(1,2,1/2)<{\cal {A}}(1,1,1)$ .

In the case of convex bodies, the quantities $\Psi _{\omega }(K)$ , $\Phi _{\omega }(K)$ are uniformly bounded by a constant that depends on $\omega $ only. We will use the following well-known consequence of the celebrated “existence of M-ellipsoids” by V. D. Milman [Reference Milman40].

Theorem 4.3 Let K be an origin-symmetric convex body in ${\mathbb R}^{n}$ . Then there exists an ellipsoid ${\cal {E}}$ such that $|{\cal {E}}|^{1/n} \leq e^{c} |K|^{1/n}$ and for every $F\in G_{n,k}$ ,

(59) $$ \begin{align} | P_{F} {\cal{E}}| \leq | P_{F} K | \leq e^{cn} | P_{F} {\cal{E}}| \end{align} $$

and

(60) $$ \begin{align} | {\cal{E}} \cap F| \leq | K \cap F | \leq e^{cn} | {\cal{E}} \cap F|, \end{align} $$

where $c>0$ is an absolute constant.

Corollary 4.4 Let $\omega \in \Pi _{n}$ such that $\omega (n)\neq n$ . Let $ \delta _{\omega } (j) := \omega (j) - \omega (j+1) + 1$ . Set

$$ \begin{align*}I_{\omega} := \{ j\leq n : \delta_{\omega} ( j) \geq 0 \} \ \mathrm{and } \ \Delta(\omega):=\frac{\min\{ \sum_{j\in I_{\omega} } \delta_{\omega}(j) , \sum_{j\in I_{\omega}^{c} } |\delta_{\omega}(j)|\} }{n - \omega(n)} + 1 .\end{align*} $$

We have that

(61) $$ \begin{align} e^{ -c \Delta( \omega) } c_{\omega} | K|^{\frac{1}{n}} \leq \Psi_{\omega} (K) \leq e^{ c \Delta( \omega) } c_{\omega} | K|^{\frac{1}{n}} \end{align} $$

and

(62) $$ \begin{align} e^{ - c \Delta( \omega)} c_{\omega} | K|^{\frac{1}{n}} \leq \Phi_{\omega} (K) \leq e^{ c \Delta( \omega)} c_{\omega} | K|^{\frac{1}{n}}, \end{align} $$

where $c>0$ is an absolute constant.

Proof Set $ \Delta _{+}(\omega ) := \frac { \sum _{j\in I_{\omega } } \delta _{\omega }(j) }{n - \omega (n)}$ and $ \Delta _{-} ( \omega ):= \frac { \sum _{j\in I_{\omega }^{c}} |\delta _{\omega }(j) |}{n - \omega (n)}$ . In the ensuing computation, we use the bounds on $|K\cap F|$ in (60): for $j\in I_{\omega }$ , we use the upper bound, while for $j \notin I_{\omega }$ , we use the lower bound. Thus, we obtain

$$ \begin{align*} \Psi_{\omega} (K) &= \left( \int_{F_{n}} \prod_{j=1}^{n-1} | K \cap F_{j}|^{ \omega(j) - \omega(j+1)+1} d \xi \right)^{\frac{1}{n(n- \omega(n))}}\\ &\leq \left( \int_{F_{n}} e^{cn \sum_{j\in I_{\omega}} (\omega(j) - \omega(j+1) +1)} \prod_{j=1}^{n-1} | {\cal{E}} \cap F_{j}|^{ \omega(j) - \omega(j+1)+1} d \xi \right)^{\frac{1}{ n(n- \omega(n))}} \\ & = e^{ c\Delta_{+}(\omega) } \left( \int_{F_{n}} \prod_{j=1}^{n-1} | {\cal{E}} \cap F_{j}|^{ \omega(j) - \omega(j+1)+1} d \xi \right)^{\frac{1}{ n(n- \omega(n))}}\\ &=e^{ c\Delta_{+}(\omega) }c_{\omega} | {\cal{E}}|^{\frac{1}{n}} \\ &\leq e^{ c ( \Delta_{+}( \omega) + 1 )}c_{\omega} | K |^{\frac{1}{ n}}. \end{align*} $$

One can verify that a similar inequality with the quantity $ \Delta _{-} (\omega )$ holds as well, which leads to the right-hand side in (61). The proof of the other inequalities is identical and hence is omitted.

5 Functional forms

In this section, we derive functional forms of some of the previous geometric inequalities. Section 5.1 concerns the dual setting with sharp inequalities for restrictions of functions to flags of subspaces; these do not depend on their counterparts for sets. In Section 5.2, we define a new notion of $\mathbf {r}$ -flag affine quermassintegrals for functions and establish a double-sided approximate isoperimetric inequality.

5.1 Functional forms of dual $\mathbf {r}$ -flag affine quermassintegrals

Let $ f $ be a bounded integrable function on $\mathbb {R}^{n}$ . We denote by $I(f)$ the functional form of the dual $\mathbf {r}$ -flag affine quermassintegral

$$ \begin{align*}I(f) := \int_{ F_{\mathbf{r}}^n } \prod_{j=1}^{r} \| f |_{F_j}\|^{i_{j+1}-i_{j-1}}_{1} d\xi_{\mathbf{r}} .\end{align*} $$

Theorem 5.1 For every $ g \in SL_{n}$ , $ I( g\cdot f) = I( f)$ , where $g\cdot f(x)=f(g^{-1}x)$ .

Proof Starting with the left-hand side, $I( g\cdot f)$ , we do a global change of variables (21) on the flag manifold:

$$ \begin{align*} I( g\cdot f) = \int_{ F_{\mathbf{r}}^n } \prod_{j=1}^{r} \| g\cdot f |_{F_j}\|^{i_{j+1}-i_{j-1}}_{1} d\xi_{\mathbf{r}} = \int_{ F_{\mathbf{r}}^n } \prod_{j=1}^{r} \| g\cdot f |_{g\cdot F_j}\|^{i_{j+1}-i_{j-1}}_{1} \sigma_{F_{\mathbf{r}}^{n} }(g, \xi) d\xi_{\mathbf{r}}, \end{align*} $$

where by (23) $\sigma _{F_{\mathbf {r}}^{n} }(g, \xi ) = \sigma _{i_{1}}^{-i_{2}}(g, F_1) \sigma _{i_{2}}^{i_{1}-i_{3}}(g, F_2) \cdots \sigma _{i_{r}}^{i_{r-1}-n}(g, F_r) $ . Now we do r local changes of variables (20) on each nested subspace $F_j$ in the product. For each $1\leq j \leq r$ , we thus have

$$ \begin{align*}\| g\cdot f|_{g\cdot F_j}\|_{1}=\|f|_{F_j}\|_{1} \, \sigma_{i_j}(g,F_j).\end{align*} $$

For the product under the integral, we obtain

$$ \begin{align*}\prod_{j=1}^{r} \| g\cdot f |_{g\cdot F_j}\|^{i_{j+1}-i_{j-1}}_{1} = \prod_{j=1}^{r} \left( \|f|_{F_j}\|_{1} \, \sigma_{i_j}(g,F_j) \right)^{i_{j+1}-i_{j-1}} = \prod_{j=1}^{r} \|f|_{F_j}\|^{i_{j+1}-i_{j-1}}_{1} \sigma_{F_{\mathbf{r}}^{n} }^{-1}(g, \xi).\end{align*} $$

The latter result admits several generalizations as in [Reference Dann, Paouris and Pivovarov12] for functional forms of dual quermassintegrals. Rather than taking $L_1(F_j)$ norms, one can take $L_{p_j}(F_j)$ norms and replace the powers $i_{j+1}-i_{j-1}$ by $\alpha _j$ . As long as $\frac {\alpha _j}{p_j}=i_{j+1}-i_{j-1}$ and the integrals exist, the conclusion of Theorem 5.1 will hold. Theorem 5.1 also generalizes to a product of m functions. One can replace $ \| f |_{F_j}\|^{i_{j+1}-i_{j-1}}_{1} $ by $ \prod _{i=1}^m \| f_i |_{E_j} \|^{\alpha _{i,j}}_{p_{i,j}} $ . For the analogue of Theorem 5.1 to hold in this case, we have to require $\sum _{i=1}^m \frac {\alpha _{i,j}}{p_{i,j}} = i_{j+1}-i_{j-1}$ . Another way to generalize functional forms of dual quermassintegrals is to replace

$$ \begin{align*}\| f |_{F_j}\|^{i_{j+1}-i_{j-1}}_{1} \quad \text{ by } \quad \frac{ \| f |_{F_j} \|^{\alpha_j}_{p_j}}{{\| g |_{F_j}\|^{\beta_j}_{q_j}}} \quad \text{ with } \quad \frac{\alpha_j}{p_j} - \frac{\beta_j}{q_j} = i_{j+1}-i_{j-1},\end{align*} $$

to ensure they remain invariant under volume-preserving transformations. Letting $q_j \to \infty $ modifies the integrand to $\frac { \| f |_{F_j} \|^{\alpha _j}_{p_j}}{{\| f |_{F_j}\|^{\beta _j}_{\infty }}}$ and the condition on the powers and norms to $\frac {\alpha _j}{p_j} = i_{j+1}-i_{j-1} $ . Note that in this case the invariance holds for arbitrary powers $\beta _j$ . As a particular case, this proves invariance under volume-preserving transformations of the integrand appearing in the next theorem. One can also take the quotient of products of functions, replacing

$$ \begin{align*}\| f |_{F_j}\|^{i_{j+1}-i_{j-1}}_{1} \quad \text{ by } \quad \frac{ \prod_{i=1}^m \| f_i |_{F_j} \|^{\alpha_{i,j}}_{p_{i,j}} }{ \prod_{l=1}^{m^{\prime}} \| g_l |_{F_j} \|^{\beta_{l,j}}_{q_{l,j}} } \quad \text{ with } \quad \sum_{i=1}^m \frac{\alpha_{i,j}}{p_{i,j}} - \sum_{l=1}^{m^{\prime}} \frac{\beta_{l,j}}{q_{l,j}}=i_{j+1}-i_{j-1}.\end{align*} $$

Here again, we can let $q_{l,j} \rightarrow \infty $ , obtaining the corresponding generalization with no restrictions on $\beta _{l,j}$ .

Theorem 5.2 Let f be a non-negative bounded integrable function on $\mathbb R^n$ , then

$$ \begin{align*}\int_{F_{\mathbf{r}}^n} \prod_{j=1}^r \frac{\|f|_{F_j}\|^{i_{j+1}-i_{j-1}}_1}{\|f|_{F_j}\|^{i_{j+1}-i_{j}}_{\infty}} d \xi_{\mathbf{r}} \leq \prod_{j=1}^r \frac{\omega_{i_j}^{i_{j+1}}}{\omega_{i_{j+1}}^{i_j}} \|f\|_1^{i_r}.\end{align*} $$

Proof The result follows by iteration of an inequality on $G_{n,k}$ for one function from our previous work [Reference Dann, Paouris and Pivovarov12, Theorem 1.2]:

(63) $$ \begin{align} \int_{G_{n,k}}\frac{\|f|_{E}\|^{n}_1}{\|f|_{E}\|^{n-k}_{\infty}} dE \leq \frac{\omega_k^{n}}{\omega_n^k}\|f\|^k_1. \end{align} $$

Applying the latter inequality repeatedly, we get

$$ \begin{align*} \int_{F_{\mathbf{r}}^n} &\prod_{j=1}^r \frac{\|f|_{F_j}\|^{i_{j+1}-i_{j-1}}_1}{\|f|_{F_j}\|^{i_{j+1}-i_{j}}_{\infty}} d \xi_{\mathbf{r}} \\ &= \int_{G_{n,i_r}} \ldots \int_{G_{F_3,i_2}} \prod_{j=2}^r \frac{\|f|_{F_j}\|^{i_{j+1}-i_{j-1}}_1}{\|f|_{F_j}\|^{i_{j+1}-i_{j}}_{\infty}} \int_{G_{F_2,i_1}} \frac{\|f|_{F_1}\|^{i_{2}}_1}{\|f|_{F_1}\|^{i_{2}-i_{1}}_{\infty}} dF_1 dF_2 \ldots dF_r\\ &\leq \frac{\omega_{i_1}^{i_2}}{\omega^{i_1}_{i_2}} \int_{G_{n,i_r}} \ldots \int_{G_{F_3,i_2}} \prod_{j=2}^r \frac{\|f|_{F_j}\|^{i_{j+1}-i_{j-1}}_1}{\|f|_{F_j}\|^{i_{j+1}-i_{j}}_{\infty}} \|f|_{F_2}\|^{i_{1}}_1 dF_2 \ldots dF_r \\ &= \frac{\omega_{i_1}^{i_2}}{\omega^{i_1}_{i_2}} \int_{G_{n,i_r}} \ldots \int_{G_{F_4,i_3}} \prod_{j=3}^r \frac{\|f|_{F_j}\|^{i_{j+1}-i_{j-1}}_1}{\|f|_{F_j}\|^{i_{j+1}-i_{j}}_{\infty}} \int_{G_{F_3,i_2}} \frac{\|f|_{F_2}\|^{i_{3}}_1}{\|f|_{F_2}\|^{i_{3}-i_{2}}_{\infty}} dF_2 dF_3 \ldots dF_r \\ &\leq \frac{\omega_{i_1}^{i_2}}{\omega^{i_1}_{i_2}} \frac{\omega_{i_2}^{i_3}}{\omega^{i_2}_{i_3}} \int_{G_{n,i_r}} \ldots \int_{G_{F_4,i_3}} \prod_{j=3}^r \frac{\|f|_{F_j\|^{i_{j+1}-i_{j-1}}_1}}{\|f|_{F_j}\|^{i_{j+1}-i_{j}}_{\infty}} \|f|_{F_3}\|^{i_{2}}_1 dF_3 \ldots dF_r\\ &=\cdots \\ &\leq \prod_{j=1}^r \frac{\omega_{i_j}^{i_{j+1}}}{\omega_{i_{j+1}}^{i_j}} \|f\|_1^{i_r}.\\[-34pt] \end{align*} $$

In [Reference Dann, Paouris and Pivovarov12], more general versions of (63) are proved with several functions and different powers. These also carry over to extremal inequalities on flag manifolds by mimicking the previous proof. As a sample, we mention just one statement. Let $1 \leq q \leq i_1$ and let $f_1, \ldots , f_q$ be non-negative bounded integrable functions on ${\mathbb R}^n$ , then

$$ \begin{align*} \int_{F_{\mathbf{r }}^n} \prod_{k=1}^q \prod_{j=2}^r \frac{\|f_k|_{F_j}\|^{\frac{i_{j+1}-i_{j}}{i_j}}_1}{\|f_k|_{F_j}\|^{\frac{i_{j+1}-i_{j}}{i_j}}_{\infty}} \frac{\left\lVert f_k|_{F_1}\right\rVert_1^{\frac{i_2}{i_1}}}{\left\lVert f_k|_{F_1}\right\rVert^{\frac{i_2-i_1}{i_1}}_{\infty}} d \xi_{\mathbf{r}} \leq \prod_{j=1}^r \frac{\omega_{i_j}^{i_{j+1}} }{\omega_{i_{j+1}}^{i_j}} \prod_{k=1}^q \|f_k\|_1. \end{align*} $$

5.2 Functional forms of the $\mathbf {r}$ -flag affine quermassintegrals

In this subsection, we extend the notion of $\mathbf {r}$ -flag affine quermassintegrals to functions. In particular, this will lead to functional versions of affine quermassintegrals. This is motivated by recent work of Bobkov, Colesanti, and Fragalá [Reference Bobkov, Colesanti and Fragalá4] and V. Milman and Rotem [Reference Milman and Rotem39]. The latter authors proposed and studied a notion of quermassintegrals for $\log $ -concave or even quasi-concave functions, which we now recall.

Definition 5.3 Suppose that $f:\mathbb {R}^n\rightarrow [0,\infty )$ is upper-semicontinuous and quasi-concave. For $ 1\leq k \leq n$ , let

$$ \begin{align*} V_k (f) := \int_{0}^{\infty} V_k( \{ f\geq t\}) d t. \end{align*} $$

The above definition is consistent with the notion of projection of a function onto a subspace as introduced by Klartag and V. Milman in [Reference Klartag and Milman25]. Namely, let $f:\mathbb R^{n}\rightarrow [0,\infty ] $ be a non-negative function and $F\in G_{n,k}$ . Define the orthogonal projection of f onto F as the function $P_{F} f: F\rightarrow [0,\infty ]$ given by

$$ \begin{align*} (P_{F} f) ( z) := \sup_{y\in F^{\perp}} f ( z+y). \end{align*} $$

Note that if K is compact and $ f:= \mathbf {1}_{K}$ , then $P_{F}f := \mathbf {1}_{P_{F}(K)}$ . Moreover, from the definition, one has

$$ \begin{align*} \{ z\in F : (P_{F}f)(z)>t \} = P_{F} ( \{ x\in {\mathbb R}^{n}: f(x) >t\}). \end{align*} $$

Assume from now on that $f:{\mathbb R}^{n} \rightarrow [0,\infty )$ and

(64) $$ \begin{align} \{x\in \mathbb{R}^n:f(x)\geq t\} \text{ is a compact body for each } t>0. \end{align} $$

For $1\leq k \leq n-1$ , we define the affine quermassintegral of f by

(65) $$ \begin{align} \Phi_{[k]} (f) := \int_{0}^{\infty} \Phi_{[k]} ( \{ f\geq t\}) dt = \int_{0}^{\infty} \left(\int_{G_{n,k}} | \{ P_{F} f\geq t\} |^{-n} d F \right)^{-\frac{1}{ nk}} dt. \end{align} $$

Note that for $f=\mathbf {1}_K$ , we have $\Phi _{[k]}(\mathbf {1}_K) = \int _{0}^1 \Phi _{[k]}(K)dt= \Phi _{[k]}(K)$ ; hence, the notions coincide for sets. For $1\leq r\leq n-1$ and $ 1\leq i_{1} < i_{2} < \cdots < i_{r} \leq n-1$ , $\mathbf {r}:= (i_{1}, \cdots , i_{r})$ , we define the $\mathbf {r}$ -flag affine quermassintegrals of f by

(66) $$ \begin{align} \Phi_{\mathbf{r}} (f) := \int_{0}^{\infty} \Phi_{\mathbf{r}} ( \{ f\geq t\}) dt. \end{align} $$

For comparison, we recall that for every $f:{\mathbb R}^{n} \rightarrow [0,\infty ] $ ,

$$ \begin{align*} \int_{{\mathbb R}^{n}} f (x) d x = \int_{0}^{\infty} | \{ f\geq t \}| dt. \end{align*} $$

For $\lambda \in {\mathbb R}\setminus \{0\}$ and f as above, we write

$$ \begin{align*} f_{(\lambda)} :{\mathbb R}^{n} \rightarrow [0,\infty] , \ \mathrm{as} \ f_{(\lambda)} ( x) := f \left(\frac{ x}{ \lambda}\right) , \end{align*} $$

and if $T\in GL_{n}$ ,

$$ \begin{align*} f\circ T :{\mathbb R}^{n} \rightarrow [0,\infty] , \ \mathrm{as} \ f\circ T ( x) := f (T^{-1} x). \end{align*} $$

Note that if $f:= \mathbf {1}_{K}$ , then

$$ \begin{align*}f_{(\lambda)} (x) = \mathbf{1}_{\lambda K} (x) \ \mathrm{and} \ f\circ T (x) = \mathbf{1}_{TK} (x) .\end{align*} $$

For $ f:{\mathbb R}^{n} \rightarrow [0, \infty ]$ , $\lambda>0$ , and $ T\in GL_{n}$ , we have

$$ \begin{align*} \{ f\circ T \geq t \} = T\left( \{ f\geq t \} \right) \ \mathrm{and } \ \{ f_{(\lambda)} \geq t \} = \lambda \{ f \geq t \}. \end{align*} $$

Thus, by the $1$ -homogeneity of the $\mathbf {r}$ -flag affine quermassintegrals for sets, and the affine invariance of these quantities, we obtain the following theorem.

Theorem 5.4 Let $f:{\mathbb R}^{n}\rightarrow [0,\infty ]$ satisfy (64). Let $\lambda>0$ and T be an affine volume-preserving map. Then

$$ \begin{align*} \Phi_{\mathbf{r}} (f_{(\lambda)} ) = \lambda \Phi_{\mathbf{r}}(f) \ \mathrm{and} \ \Phi_{\mathbf{r}} (f\circ T) = \Phi_{\mathbf{r}}(f). \end{align*} $$

We recall the symmetric decreasing rearrangement of an integrable function f. For a set $A\subseteq {\mathbb R}^{n}$ with finite volume, the decreasing rearrangement $A^{\ast }$ is defined as $ A^{\ast } := r_{A} B_{2}^{n},$ where $r_A$ is the volume-radius of A. The symmetric decreasing rearrangement $f^{\ast }$ of f is defined as the radial function $f^{\ast }$ such that, for $t>0$ ,

$$ \begin{align*}\{f\geq t\}^{\ast} = \{ f^{\ast} \geq t \}.\end{align*} $$

In particular, $r_{\{ f\geq t\} } B_{2}^{n} = \{ f^{\ast } \geq t \}$ . Using (66) and (35), we have, for non-negative quasi-concave functions f on ${\mathbb R}^{n}$ ,

$$ \begin{align*}\Phi_{\mathbf{r}} (f) = \int_{0}^{\infty} \Phi_{\mathbf{r}}(\{ f\geq t\}) d t \geq \int_{0}^{\infty} \Phi_{\mathbf{r}}(r_{\{ f\geq t\}} B_{2}^{n}) d t = \int_{0}^{\infty} \Phi_{\mathbf{r}}(\{ f^{\ast} \geq t \} ) d t= \Phi_{\mathbf{r}} (f^{\ast}) .\end{align*} $$

Lastly, let f be a non-negative quasi-concave function on $\mathbb R^{n}$ satisfying (64). We define

$$ \begin{align*}d_{BM} ( f) := \sup_{t>0}d_{BM} ( \{ f\geq t\} ).\end{align*} $$

The results of Section 3 lead to the following double-sided inequality for $\Phi _{\mathbf {r}}(f)$ :

Theorem 5.5 Let f be a non-negative quasi-concave function on ${\mathbb R}^{n}$ satisfying (64). Then

$$ \begin{align*} \Phi_{\mathbf{r}} (f^{\ast} ) \leq \Phi_{\mathbf{r}} ( f) \leq c^{\prime} \min\left\{ \sqrt{ \frac{n}{i_r} }, \log( 1+d_{BM}(f)) \right\} \Phi_{\mathbf{r}} (f^{\ast} ). \end{align*} $$