A kinematic formula for affine diameters and affine medians of a convex set
type: Item
instance of: scholarly article
title: A kinematic formula for affine diameters and affine medians of a convex set
zbMATH Open document ID: 1078.52504
DOI: 10.1023/B:JOTH.0000008767.34024.40
author: V. V. Makeev
published in: Journal of Mathematical Sciences (New York)
publication date: 2005-06-23T00:00:00Z
review text: The author considers the area measure \(S(K_i)\) resp. \(S(K_i')\) of all points of a compact convex set \(K\) in the plane \(R^2\) with \(C^2_+\)-boundary which lie on precisely \(i\) chords of \(K\) with parallel tangent lines at the ends (affine diameters) resp. \(i\) chords of \(K\) halving the area of \(K\) (affine medians). By topological (Sard's lemma) and integral geometric methods he proves 1) \(S(K_2)= S(K_4)=\cdots= S(K_\infty)= 0\) and 2) \({S(\Delta K)\over 4}\leq S(K_1)+ 3S(K_3)+ 5S(K_5)+\cdots\leq {S(\Delta K)\over 2}\) resp. \(1')\) \(S(K_2')= S(K_4')=\cdots= S(K_\infty')= 0\) and \(2')\) \(S(K_1')+ 3S(K_3')+ 5S(K_5')+\cdots\leq {S(\Delta K)\over 4}\). Hereby \(S(\Delta K)\) denotes the area of the difference body \(\Delta K\) of \(K\). Some corollaries follow. Remark: 1) also holds in \(n\) dimensions.
Mathematics Subject Classification ID: 52A38
Mathematics Subject Classification ID: 52A22
Mathematics Subject Classification ID: 53C65
Mathematics Subject Classification ID: 53A17
zbMATH DE Number: 2182773
zbMATH Keywords: affine diameter
zbMATH Keywords: affine median
zbMATH Keywords: difference body
reviewed by: Kurt Leichtweiß
MaRDI profile type: Publication
full work available at URL: b:joth.0000008767.34024.40
OpenAlex ID: W2085656589
description: scientific article; zbMATH DE number 2182773