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# Convex Bodies of Minimal Volume, Surface Area and Mean Width with Respect to Thin Shells

Published:2008-02-01
Printed: Feb 2008
• Károly Böröczky
• Károly J. Böröczky
• Carsten Schütt
• Gergely Wintsche
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## Abstract

Given $r>1$, we consider convex bodies in $\E^n$ which contain a fixed unit ball, and whose extreme points are of distance at least $r$ from the centre of the unit ball, and we investigate how well these convex bodies approximate the unit ball in terms of volume, surface area and mean width. As $r$ tends to one, we prove asymptotic formulae for the error of the approximation, and provide good estimates on the involved constants depending on the dimension.
 MSC Classifications: 52A27 - Approximation by convex sets 52A40 - Inequalities and extremum problems