A simple proof of an approximation theorem of H. Minkowski
type: Item
instance of: scholarly article
title: A simple proof of an approximation theorem of H. Minkowski
zbMATH Open document ID: 0790.52002
DOI: 10.1007/BF01264075
author: Michael Schmuckenschlaeger
published in: Geometriae Dedicata
publication date: 1994-07-07T00:00:00Z
review text: A theorem of H. Minkowski states that every convex body in \(\mathbb{R}^ n\) can be approximated by a sequence of convex bodies whose boundaries are analytic \((n-1)\)-dimensional manifolds. The author gives a simple proof of the following stronger theorem. Theorem 1.1. Let \(C\) be a convex body in \(\mathbb{R}^ n\) with polar \(C^*\). Then there exists a sequence \(C_ k\) of convex bodies with the following properties: (1) \(C_ k\downarrow C\), (2) \(\partial C_ k\) and \(\partial C_ k^*\) are analytic \((n-1)\)- dimensional manifolds, (3) The curvatures of \(\partial c_ k\) and \(\partial C_ k^*\) never vanish. He also proves some further properties of \(C_ k\).
reviewed by: Maria Moszyńska
Mathematics Subject Classification ID: 52A27
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 52A99
zbMATH DE Number: 493319
zbMATH Keywords: approximation
zbMATH Keywords: analytic convex body
zbMATH Keywords: curvature
zbMATH Keywords: polar
MaRDI profile type: Publication
cites work: Q4753788
cites work: Q5547252
cites work: Beliebige konvexe Polytope als Schnitte und Projektionen höherdimensionaler Würfel, Simplizes und Masspolytope
cites work: Optimization of functions on certain subsets of Banach spaces
cites work: Smooth approximation of convex bodies
full work available at URL: https://doi.org/10.1007/bf01264075
OpenAlex ID: W2055208512
description: scientific article; zbMATH DE number 493319