Convex polytopes whose projection bodies and difference sets are polars
type: Item
instance of: scholarly article
title: Convex polytopes whose projection bodies and difference sets are polars
zbMATH Open document ID: 0679.52002
DOI: 10.1007/BF02574676
author: Horst Martini
published in: Discrete \& Computational Geometry
publication date: 1991-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/131145
review text: By means of analytic methods and combinatorial properties of hyperplane arrangements it is shown that a convex d-polytope P (d\(\geq 3)\) is a simplex if and only if its projection body \(\Pi\) P and its difference body DP are polars. For \(d=2\) this polarity holds precisely for those convex figures whose difference sets are bounded by Radon curves, for \(d\geq 3\) (besides simplices) obviously the ellipsoids have this correspondence. Thus, there remains the question whether simplices and ellipsoids are the only convex bodies in \({\mathbb{R}}^ d\) (d\(\geq 3)\) with this property. At least it is known that polarity of \(\Pi\) K and DK (with respect to a sphere of radius \(r\in {\mathbb{R}}^+)\) for an arbitrary convex body K of volume V(K) implies \[ (2\omega_{d-1}/\omega_ d)V(K)\leq r^ 2\leq d V(K),\quad \omega_ d:=\pi^{d/2}/\Gamma (1+d/2),\quad d\geq 2, \] with equality on the left hand side for ellipsoids and on the right hand side if and only if K is a simplex.
reviewed by: Horst Martini
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 05B45
zbMATH DE Number: 4113411
zbMATH Keywords: ellipsoid
zbMATH Keywords: convex d-polytope
zbMATH Keywords: simplex
zbMATH Keywords: projection body
zbMATH Keywords: difference body
zbMATH Keywords: polarity
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4113411