The average number of normals through a point in a convex body and a related Euler-type identity
type: Item
instance of: scholarly article
title: The average number of normals through a point in a convex body and a related Euler-type identity
zbMATH Open document ID: 0793.52001
DOI: 10.1007/BF01265675
author: Kathy Hann
published in: Geometriae Dedicata
publication date: 1994-02-01T00:00:00Z
review text: A normal to a convex body \(K\) at a point \(x\) in the boundary of \(K\) is a ray with end point at \(x\), perpendicular to a support plane \(H\) of \(K\) at \(x\), and contained in the halfspace bounded by \(H\) that contains \(K\). For each point \(p\) in \(K\) let \(n(K,p)\) be the number of normals to \(K\) passing through \(p\). Then the average number of normals through a point in \(K\) is defined by \[ n(K)={I(K)\over V(K)} \quad \text{where} \quad I(K)=\int_ K n(K,p)dV, \] \(V(K)\) denoting the volume of \(K\). The problem to find bounds for \(n(K)\) has been solved completely by \textit{G. D. Chakerian} [`The number of diameters through a point inside an oval', Unión Matemática Argentina 29, 282-290 (1984)] by showing \(2\leq n(K)\leq 2\pi/(\pi-\sqrt 3)\) with equality holding in the upper bound only for the Reuleaux triangle and in the lower bound only for a circular disk. In this paper bounds for more general cases are obtained. In the planar case the author shows for convex domains being bounded by a sufficiently smooth convex curve or by a polygon that \(n(K)\leq 12\) (in general), \(n(K)\leq 8\) (centrally symmetric), \(n(K)\leq 6\) (centers of curvature inside K). We have \(n(K)=8\) for regular \(2n\)-gons. In higher dimensions \(m\) she proves \(n(K)\leq{V(K+DK)\over V(K)}-1\) (sufficiently smooth or polytope) and \(n(K)\leq 3^ m-1\) (centrally symmetric in addition) where \(DK\) denotes the difference body of \(K\); \(n(K)=3^ m-1\) for the \(m\)- cube. Finally an Euler type relation is studied in general which emanated from the proof of the bound for polygons.
Mathematics Subject Classification ID: 52A10
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 52A38
zbMATH DE Number: 486898
zbMATH Keywords: centrally symmetric
zbMATH Keywords: normal
zbMATH Keywords: convex body
zbMATH Keywords: average number
zbMATH Keywords: Euler type relation
reviewed by: Bernd Wegner
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01265675
OpenAlex ID: W2330685373
description: scientific article; zbMATH DE number 486898