Contraction of convex hypersurfaces by their affine normal
type: Item
instance of: scholarly article
title: Contraction of convex hypersurfaces by their affine normal
zbMATH Open document ID: 0858.53005
author: Ben Andrews
published in: Journal of Differential Geometry
publication date: 1997-03-10T00:00:00Z
review text: The author considers an affine invariant evolution equation for convex hypersurfaces in Euclidean space by assigning to each point a velocity equal to the unimodular affine normal vector. For an arbitrary compact, smooth, strictly convex initial hypersurface, it is shown that this deformation produces a unique, smooth family of convex hypersurfaces which converge to a point in finite time. Moreover, it is also shown that the hypersurfaces converge smoothly to an ellipsoid after rescaling about the final point so to make the enclosed volume constant. Finally, the author uses this result in order to give new proofs of two known affine geometric consequences: the Blaschke-Santaló-Deicke affine isoperimetric inequality and the so-called Blaschke-Santaló inequality.
Mathematics Subject Classification ID: 53A15
Mathematics Subject Classification ID: 35G10
Mathematics Subject Classification ID: 35K25
Mathematics Subject Classification ID: 58J35
Mathematics Subject Classification ID: 53C21
zbMATH DE Number: 895024
zbMATH Keywords: geometric inequalities
zbMATH Keywords: affine invariant evolution equation
zbMATH Keywords: convex hypersurfaces
zbMATH Keywords: Euclidean space
zbMATH Keywords: affine isoperimetric inequality
reviewed by: Salvador D. Gigena
MaRDI profile type: Publication
Wikidata QID: Q115177758
full work available at URL: https://doi.org/10.4310/jdg/1214458106
OpenAlex ID: W1580042292
DOI: 10.4310/JDG/1214458106
description: scientific article; zbMATH DE number 895024