The purpose of this note is to establish a new version of the local Steiner formula and to give an application to convex bodies of constant width. This variant of the Steiner formula generalizes results of Hann [3] and Hug [6], who use much less elementary techniques than the methods of this paper. In fact, Hann asked for a simpler proof of these results [4, Problem 2, p. 900]. We remark that our formula can be considered as a Euclidean analogue of a spherical result proved in [2, p. 46], and that our method can also be applied in hyperbolic space.

For some remarks on related formulas in certain two-dimensional Minkowski spaces, see Hann [5, p. 363].

For further information about the notions used below, we refer to Schneider's book [9]. Let [Kscr ]n be the set of all convex bodies in Euclidean space IRn, that is, the set of all compact, convex, non-empty subsets of IRn. Let Sn-1 be the unit sphere. For K∈[Kscr ]n, let NorK be the set of all support elements of K, that is, the pairs (x, u)∈IRn×Sn−1 such that x is a boundary point of K and u is an outer unit normal vector of K at the point x. The support measures (or generalized curvature measures) of K, denoted by Θ0(K, ·), …, Θn−1(K, ·), are the unique Borel measures on IRn×Sn−1 that are concentrated on NorK and satisfy

formula here

for all integrable functions f[ratio ]IRn→IR; here λ denotes the Lebesgue measure on IRn. Equation (1), which is a consequence and a slight generalization of Theorem 4.2.1 in Schneider [9], is called the local Steiner formula. Our main result is the following.