# Relationships between Widths of a Convex Body and of an Inscribed Parallelotope

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Marek Lassak

Institut für Informatik

Freie Universität Berlin

Takustr. 9, D-14195 Berlin

email: lassak@inf.fu-berlin.de

Report B 00-08

April 2000

**Abstract**

Assume that a parallelotope *P* is inscribed in a three-dimensional convex
body *C*. A conjecture says that *w*^{-1}_{1}+w^{-1}_{2}+w^{-1}_{3}>=1, where *w*_{i}
is the ratio of the width of *C* to the width of *P* for the
direction perpendicular to the *i*-th pair of parallel facets of *P*. We prove three weaker inequalities. One of them is *w*^{-1}_{1}+w^{-1}_{2}+a^{-1}_{3}>=1, where *a*_{3} denotes the related axial diameter of *C*.

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