Mixed affine surface area
type: Item
instance of: scholarly article
title: Mixed affine surface area
zbMATH Open document ID: 0632.52005
DOI: 10.1016/0022-247X(87)90097-7
author: Erwin Lutwak
published in: Journal of Mathematical Analysis and Applications
publication date: 1987-01-01T00:00:00Z
review text: The author introduces the mixed affine area of convex bodies \(K_ 1,...,K_ n\) in \(E^ n\) having continuous curvature functions \(f(K_ i,\cdot)\) \((i=1,...,n)\) in the sense of Minkowski by means of \[ \Omega (K_ 1,...,K_ n)=\int_{S^{n-1}}[f(K_ 1,u)...f(K_ n,u)]^{1/(n+1)}dS(u), \] where \(S^{n-1}\) is the unit sphere and dS its area element. It the boundary of K is of class \(C^ 2\), then f(K,\(\cdot)\) is the reciprocal Gauss curvature as a function of the exterior unit normal vector, hence \(\Omega\) (K,...,K) is the classical affine area of K. In this paper, the mixed affine area is related to various other functionals of convex bodies, and known inequalities are combined to yield several new inequalities involving the mixed affine area. Example: \[ \Omega (K_ 1,...,K_ n)^{n+1}\leq n^{n+1} \omega^ 2_ n V(K_ 1,...,K_ n)^{n-1} \] (\(\omega\) \({}_ n=volume\) of the unit ball in \(E^ n)\) with equality if and only if the \(K_ i\) are homothetic ellipsoids.
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 53A15
zbMATH DE Number: 4028288
zbMATH Keywords: mixed volume
zbMATH Keywords: mixed affine area
zbMATH Keywords: curvature functions
reviewed by: Rolf Schneider
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4028288