Convex bodies with extremal volumes having prescribed brightness in finitely many directions
type: Item
instance of: scholarly article
title: Convex bodies with extremal volumes having prescribed brightness in finitely many directions
zbMATH Open document ID: 0838.52011
DOI: 10.1007/BF01264932
author: Stefano Campi
author: A. Colesanti
author: Paolo Gronchi
published in: Geometriae Dedicata
publication date: 1995-11-21T00:00:00Z
review text: Let \(K\) be a convex body in \(n\)-dimensional Euclidean space and let \(W = \{w_1, \dots, w_m\}\) be a set of unit vectors containing a basis of the space. The symbol \({\mathcal H} (K,W)\) denotes the family of convex bodies whose \((n - 1)\)-dimensional volumes of the projections onto hyperplanes orthogonal to \(w_1, \dots, w_m\) coincide with the volumes of analogical projections of \(K\). The authors prove that there exists a unique convex body (up to translation) in \({\mathcal H} (K,W)\) with maximum volume. This body is a centrally-symmetric polytope. Moreover, it is proved that every convex body from \({\mathcal H} (K,W)\) of the minimum volume is a polytope with at most \(n + m\) facets.
reviewed by: Marek Lassak
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 52A38
Mathematics Subject Classification ID: 52A39
zbMATH DE Number: 806277
zbMATH Keywords: facet
zbMATH Keywords: zonoid
zbMATH Keywords: convex body
zbMATH Keywords: volume
zbMATH Keywords: centrally-symmetric
zbMATH Keywords: polytope
MaRDI profile type: Publication
cites work: Shadows of Convex Bodies
cites work: Approximation of zonoids by zonotopes in fixed directions
cites work: Q3245768
cites work: An inequality related to the isoperimetric inequality
cites work: Q4274002
cites work: Ein allgemeines Extremalproblem für konvexe Körper
cites work: Q5657318
cites work: Q5287551
cites work: Restricted chord projection and affine inequalities
description: scientific article; zbMATH DE number 806277