On the history of the affine surface area for convex bodies
type: Item
instance of: scholarly article
title: On the history of the affine surface area for convex bodies
zbMATH Open document ID: 0765.53008
DOI: 10.1007/BF03323201
published in: Results in Mathematics
publication date: 1992-09-27T00:00:00Z
review text: For a convex body \(K\) in \(n\)-space, whose boundary \(\partial K\) has everywhere positive Gauss curvature \(H_{n-1}\), W. Blaschke showed in 1923, that the integral over \(\partial K\) of \(H_{n-1}^{1/(n+1)}\), with respect to the Euclidean surface area measure on \(\partial K\), is invariant under the group of unimodular affine transformations. Blaschke called this the affine surface area of \(\partial K\), here denoted by \(O_{\text{aff}}(\partial K)\). This paper describes a number of subsequent definitions and representations of this invariant, as well as the extent to which they are more inclusive than those of Blaschke. Here is an example due to \textit{C. Schütt} and \textit{E. Werner} [Math. Scand. 66, No. 2, 275-290 (1990; Zbl 0739.52008)]. Let \(K_ \delta\) be the convex body, which is the common part of the intersections of \(K\) with those half-spaces, which meet \(K\) in a set of volume \(V(K)-\delta\) for \(\delta>0\) and small enough. Then \(O_{\text{aff}}(\partial K)\) is \(\lim_{\delta\to 0+} (V(K)-V(K_ \delta))/\delta^{2/(n+1)}\) up to a constant depending on \(n\) alone. Here the integration of \(H_{n- 1}^{1/(n+1)}\) is in Lebesgue's sense; this is justified by the existence almost everywhere on \(\partial K\) of the needed derivatives, as shown by A. D. Aleksandrov in 1939. The result holds for all \(K\) with interior points.
Mathematics Subject Classification ID: 53A15
Mathematics Subject Classification ID: 52C07
zbMATH DE Number: 57246
zbMATH Keywords: unimodular affine transformations
zbMATH Keywords: affine surface area
author: Kurt Leichtweiß
reviewed by: William J. Firey
MaRDI profile type: Publication
cites work: Über einige Affinvarianten konvexer Bereiche
cites work: Q5186295
cites work: Bemerkungen zur Definition einer erweiterten Affinoberfläche von E. Lutwak. (Remarks on the definition of extended affine surface area given by E. Lutwak)
cites work: Extended affine surface area
cites work: Geominimal surface area
cites work: The convex floating body.
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description: scientific article; zbMATH DE number 57246