A characterization of affine surface area
type: Item
instance of: scholarly article
title: A characterization of affine surface area
zbMATH Open document ID: 0947.52003
author: Monika Ludwig
author: Matthias Reitzner
published in: Advances in Mathematics
publication date: 2000-11-02T00:00:00Z
review text: A famous theorem by H. Hadwiger says that a real valued functional \(\mu\) on the space \({\mathcal K}^d\) of convex bodies in \(\mathbb{E}^d\) equipped with the Hausdorff topology is the linear combination \[ \mu= \sum^d_{ l=0} c_lW_l\;(c_l\in \mathbb{R},\;l=0,\dots,d) \] of the quermassintegrals \(W_l\) provided that \(\mu\) is assumed to be a valuation (i.e. there is \(\mu(K_1 \cup K_2)+ \mu(K_1\cap K_2)= \mu(K_1)+ \mu(K_2)\) for all \(K_1,K_2\in {\mathcal K}^d\) with \(K_1\cup K_2\in {\mathcal K}^d)\), motion invariant and continuous. T. Ódor posed the problem whether there exists an equiaffine analogon of Hadwiger's theorem in the following form: An equiaffine invariant and upper (resp. lower) semicontinuous valuation \(\mu:{\mathcal K}^d \to\mathbb{R}\) must be of the form \[ \mu= c_0+ c_1V+c_2\Omega\;\bigl(c_0,c_1, c_2\in\mathbb{R},\;c_1\geq 0\text{ (resp. }\leq 0) \bigr) \] where \(V\) is the volume and \(\Omega\) is the (generalized) affine surface area, defined as the Lebesgue integral of the \((d+1)\)-st root of the (a.e. existing) Gaussian curvature of the boundary of the convex body with respect to its \((d-1)\)-dimensional Hausdorff measure. The authors present a positive answer to this problem after the first author has done so in the two-dimensional case (to appear in Abh. Math. Semin. Univ. Hamburg). Their very sophisticated proof follows the basic ideas in this special case but also fully uses the boundary structure of an arbitrary convex body as known by A. D. Aleksandrov. Additionally, an induction argument and approximation by so-called \(\varepsilon\)-smooth bodies should be mentioned in this context. [Remark: As a consequence of the authors' result the equiaffine invariant valuations \(\mu_l\) with \(\mu_l(K):= \int_{\text{bd} K}H_l d\Omega\) \((H_l\) \(l\)-th affine mean curvature of \(\text{bd} K\), \(l=1,\dots, d-1)\) of sufficiently smooth convex bodies \(K\) cannot be extended to all convex bodies in a semicontinuous manner].
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 53A15
zbMATH DE Number: 1389129
zbMATH Keywords: convex bodies
zbMATH Keywords: affine surface area
zbMATH Keywords: semicontinuity
zbMATH Keywords: valuation
reviewed by: Kurt Leichtweiß
MaRDI profile type: Publication
OpenAlex ID: W1967655309
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DOI: 10.1006/AIMA.1999.1832
description: scientific article; zbMATH DE number 1389129