Valuations on convex sets of oriented hyperplanes
type: Item
instance of: scholarly article
title: Valuations on convex sets of oriented hyperplanes
zbMATH Open document ID: 1071.52006
author: John Gates
author: Daniel Hug
author: Rolf Schneider
published in: Discrete \& Computational Geometry
publication date: 2005-02-23T00:00:00Z
review text: An oriented hyperplane \(H_{u,t} := \{x \in \mathbb{R}^d \mid \langle x,u\rangle = t\}\), with \(u \in S^{d-1}\) (the unit sphere) and \(t \in \mathbb{R}\), is identified with the point \(\gamma(H_{u,t}) := u + te \in \mathbb{R}^{d+1}\), where \(e\) is a unit vector spanning the last component in \(\mathbb{R}^{d+1} = \mathbb{R}^d \times \mathbb{R}\). Such points \(\gamma(H_{u,t})\) coming from the family \({\mathcal H}^d\) of oriented hyperplanes form a cylinder \(Z = S^{d-1} \times \mathbb{R}\); further, one writes \(\nu(H_{u,t}) := u\) for the normal vector to \(H_{u,t}\). Hyperplanes \(H_{u,t}\) and \(H_{v,s}\) are antipodal if \(v = -u\). A subset \(M\) of \(Z\) which does not contain antipodal hyperplanes is convex if the cone with apex \(0\) (the origin) generated by \(M\) is convex (the situation for those \(M\) which do contain antipodal hyperplanes is more complicated); \({\mathcal K}({\mathcal H}^d)\) is the family of compact convex such subsets, and the Hausdorff metric on compact subsets of \(\mathbb{R}^{d+1}\) induces a metric on \({\mathcal K}({\mathcal H}^d)\). Recall that a mapping \(\varphi: {\mathcal S} \to \mathbb{R}\) on an intersectional class \({\mathcal S}\) of subsets of some set (that is, \(S \cap T \in {\mathcal S}\) if \(S,T \in {\mathcal S}\)) is a valuation if \(\varphi(S \cup T) + \varphi(S \cap T) = \varphi(S) + \varphi(T)\) whenever \(S,T,S \cup T \in {\mathcal S}\). One natural valuation on \({\mathcal K}({\mathcal H}^d)\) is the restriction \(\eta\) of the Haar measure. Others are the spherical analogues \(v_j\) of the intrinsic volumes on the image space \(\nu({\mathcal K}({\mathcal H}^d))\). As an analogue of the well-known characterization by Hadwiger of the intrinsic volumes, the authors conjecture that a continuous rigid motion invariant valuation \(\varphi\) on \({\mathcal K}({\mathcal H}^d)\) is of the form \[ \varphi(K) = c_d\eta(K) + \sum_{j=0}^{d-1} \, c_jv_j(\nu(K)) \] for \(K \in {\mathcal K}({\mathcal H}^d)\), with \(c_…
reviewed by: Peter McMullen
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 52A55
Mathematics Subject Classification ID: 52B45
Mathematics Subject Classification ID: 52A38
zbMATH DE Number: 2138097
zbMATH Keywords: oriented hyperplane
zbMATH Keywords: convex set
zbMATH Keywords: rigid motion invariant
zbMATH Keywords: Hadwiger's theorem
MaRDI profile type: Publication
full work available at URL: s00454-004-1126-2
OpenAlex ID: W2065733818
DOI: 10.1007/S00454-004-1126-2
description: scientific article; zbMATH DE number 2138097