On planar convex bodies of given Minkowskian thickness and least possible area
type: Item
instance of: scholarly article
title: On planar convex bodies of given Minkowskian thickness and least possible area
zbMATH Open document ID: 1077.52006
author: Gennadiy Averkov
published in: Archiv der Mathematik
publication date: 2005-04-14T00:00:00Z
review text: Let \(B\) be the unit disc in a \(2\)-dimensional normed space (Minkowski space). The {Minkowski thickness} of a convex body \(K\), \(\Delta_B(K)\) is defined to be \(\min\{ (h_K(u) + h_K(-u))/h_B(u) : | u| = 1\}\) where \(h\) denotes the support function of the indicated body and \(| \cdot| \) is an auxiliary Euclidean norm. The author investigates the problem of minimizing the area of \(K\) subject to the condition \(\Delta_B(K) = 1\) and denotes the set of minimizers by \(\chi(B)\). In the Euclidean case (with \(B=E\)) it is known [\textit{J. Pál}, Math. Ann. 83, 311--319 (1921; JFM 48.0838.02)] that \(\chi(E)\) is the set of equilateral triangles of height \(1\). The author proves that for general \(B\), (i) \(\chi(B)\) always contains triangles and the difference body of any such triangle (an affine regular hexagon) is circumscribed to \(B\); (ii) if this set does not exhaust \(\chi(B)\) then the remaining elements are quadrilaterals; and (iii) he characterizes when quadrilaterals are present in \(\chi(B)\). The question in dimension \(\geq 3\) (even for the Euclidean case) is completely open.
Mathematics Subject Classification ID: 52A21
Mathematics Subject Classification ID: 52A10
Mathematics Subject Classification ID: 52A38
zbMATH DE Number: 2155977
zbMATH Keywords: Minkowski thickness
zbMATH Keywords: minimal volume
zbMATH Keywords: Minkowski constant width
zbMATH Keywords: affine hexagons
zbMATH Keywords: equilateral triangles
MaRDI profile type: Publication
full work available at URL: s00013-004-1152-6
OpenAlex ID: W2057152887
DOI: 10.1007/S00013-004-1152-6
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description: scientific article; zbMATH DE number 2155977