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$L_p$ John Ellipsoids

Published online by Cambridge University Press:  25 February 2005

Erwin Lutwak ,
Deane Yang  and
Gaoyong Zhang
Erwin Lutwak
Affiliation:
Department of Mathematics, Polytechnic University, Brooklyn, NY 11201, USA. E-mail: elutwak@poly.edu, dyang@poly.edu, gzhang@poly.edu
Deane Yang
Affiliation:
Department of Mathematics, Polytechnic University, Brooklyn, NY 11201, USA. E-mail: elutwak@poly.edu, dyang@poly.edu, gzhang@poly.edu
Gaoyong Zhang
Affiliation:
Department of Mathematics, Polytechnic University, Brooklyn, NY 11201, USA. E-mail: elutwak@poly.edu, dyang@poly.edu, gzhang@poly.edu
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Abstract

It is shown that the classical John ellipsoid, the Petty ellipsoid and a recently discovered ‘dual’ of the Legendre ellipsoid are all special cases ($p = \infty$, $p = 1$ and $p = 2$) of a family of $L_p$ ellipsoids which can be associated with a fixed convex body. This insight allows for a unified view of, alternative approaches to, and extensions of some basic results in convex geometry.

Keywords

$L_p$-curvature$L_p$ Brunn–Minkowski theoryJohn ellipsoidPetty ellipsoidLegendre ellipsoidinertial ellipsoidisotropic positionvolume-ratio52A40
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 90 , Issue 2 , March 2005 , pp. 497 - 520
Copyright
2005 London Mathematical Society

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Footnotes

Research supported, in part, by NSF Grants DMS–9803261 and DMS–0104363.