Determination of convex bodies by projection functions
type: Item
instance of: scholarly article
title: Determination of convex bodies by projection functions
zbMATH Open document ID: 0926.52004
DOI: 10.1023/A:1005022322391
author: Christina Bauer
published in: Geometriae Dedicata
publication date: 1999-11-29T00:00:00Z
review text: The author proves some interesting results related to projection functions of convex bodies, i.e., related to the question to what extent a compact, convex set with interior points in \(\mathbb{R}^d\), \(d\geq 2\), is (almost) determined by its \(i\)th projection function defined on the Grassmannian of \(i\)-dimensional linear subspaces of \(\mathbb{R}^d\), with \(i\in \{1, \dots, d-1\}\). E.g., it is shown that a convex body \(K\) is almost determined by its brightness function (i.e., \(i=d-1)\) if and only if \(K\) is a polytope with \(f_{d-1} (K)<2d+2\) and \(f_{d-1} (\nabla K)\leq 2d+2\), where \(f_{d-1}\) denotes the number of facets and \(\nabla K\) is written for the Blaschke body of \(K\). (If the Blaschke sum \((1-\alpha)^{1/d-1} K\#\alpha^{1/d-1}K^*\), \(0\leq \alpha\leq 1\), of \(K\) and its reflection \(K^*\) at the origin as well as translates of it are the only convex bodies having the same brightness function as \(K\), then \(K\) is said to be almost determined by this function.) Further on, it is proved that a convex \(d\)-polytope, \(d\geq 4\), is determined by its \(i\)-th projection function, \(i\in\{2, \dots, d-2\}\), if all its \((i+1)\)-faces are parallelepipeds.
reviewed by: Horst Martini
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 52B11
zbMATH DE Number: 1241004
zbMATH Keywords: Blaschke body
zbMATH Keywords: irreducible polytopes
zbMATH Keywords: zonotope
zbMATH Keywords: projection functions
zbMATH Keywords: convex bodies
zbMATH Keywords: brightness function
zbMATH Keywords: Blaschke sum
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1241004