Upper semicontinuous valuations on the space of convex discs
type: Item
instance of: scholarly article
title: Upper semicontinuous valuations on the space of convex discs
zbMATH Open document ID: 0960.52002
DOI: 10.1023/A:1005240024644
author: Monika Ludwig
published in: Geometriae Dedicata
publication date: 2001-05-10T00:00:00Z
review text: A valuation on the space \({\mathcal K}^2\) of non-empty compact convex sets in the euclidean plane \(\mathbb{E}^2\) is a function \(\mu:{\mathcal K}^2\to \mathbb{R}\) such that \(\mu(K)+ \mu(L)= \mu(K\cup L)+\mu (K\cap L)\) whenever \(K,L\), \(K \cup L\in {\mathcal K}^2\). An old result of \textit{H. Hadwiger} [see Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, Springer Berlin (1957; Zbl 0078.35703)] states that such a valuation which is rigid motion invariant and continuous with respect to the Hausdorff metric on \({\mathcal K}^2\) is a linear combination of the Euler characteristic, perimeter and area of the sets in \({\mathcal K}^2\). Here, the author relaxes continuity to upper semi-continuity, with the result that a suitable curvature integral of the sets in \({\mathcal K}^2\) must be added to Hadwiger's list.
Mathematics Subject Classification ID: 52A10
Mathematics Subject Classification ID: 52A38
Mathematics Subject Classification ID: 52A39
zbMATH DE Number: 1485589
zbMATH Keywords: convex disc
zbMATH Keywords: valuation
zbMATH Keywords: rigid motion invariant
zbMATH Keywords: upper semi-continuity
reviewed by: Peter McMullen
MaRDI profile type: Publication
full work available at URL: a:1005240024644
OpenAlex ID: W199940885
description: scientific article; zbMATH DE number 1485589