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On parallelepipeds of minimal volume containing a convex symmetric body in ℝn

Published online by Cambridge University Press:  24 October 2008

A. Pelczynski  and
S. J. Szarek
A. Pelczynski
Affiliation:
Institute of Mathematics, Polish Academy of Sciences, Sniadeckich 8, Ip., 00950 Warszawa, Poland
S. J. Szarek
Affiliation:
Department of Mathematics, Case Western Reserve University, Cleveland Ohio 44106, U.S.A. Institut des Hautes Etudes Scientifiques, 91440 Bures-Sur-Yvette, France
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Abstract

Given a convex symmetric body C ⊂ ℝn we put a(C) = |C| sup |P|−1 where the supremum extends over all parallelepipeds containing C and |A| denotes the volume of a set A ⊂ ℝn. Let an = inf {a(C): C ⊂ ℝn}. We show that

which slightly improves the estimate due to Dvoretzky and Rogers [6]. In every dimension n we construct a convex symmetric polytope Wn such that the unit Euclidean ball is the ellipsoid of maximal volume inscribed into Wn and the volume of every parallelepiped containing Wn is greater than

for large n which shows ‘the limit’ to the Dvoretzky Rogers method for bounding an below. We present an alternative proof of the result of I. K. Babenko [l] that . We show that , and that a local minimum of the function C→a(C) for C ⊂ ℝn is attained only at an equiframed convex body (that is, a body such that every point of its boundary belongs to a parallelepiped of minimal volume containing the body).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 109 , Issue 1 , January 1991 , pp. 125 - 148
Copyright
Copyright © Cambridge Philosophical Society 1991

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