Does polynomial parallel volume imply convexity?
type: Item
instance of: scholarly article
title: Does polynomial parallel volume imply convexity?
zbMATH Open document ID: 1048.52003
author: Matthias Heveling
author: Daniel Hug
author: Günter Last
published in: Mathematische Annalen
publication date: 2004-07-01T00:00:00Z
review text: For a set \(A \subset \mathbb R^d\) let \(A_r\) denote the set of points with distance at most \(r\) from \(A\). The Steiner formula says that the volume of \(A_r\), \(V(A_r)\), is a polynomial in \(r\) if \(A\) is convex. This paper is about the reverse question. It is shown that if \(d=2\) and if \(V(A_r)\) is polynomial in \(r\), then \(A\) is convex. This is not true for \(d\geq 3\).
reviewed by: Imre Bárány
Mathematics Subject Classification ID: 52A38
Mathematics Subject Classification ID: 28A75
zbMATH DE Number: 2078026
zbMATH Keywords: convex bodies
zbMATH Keywords: parallel body
zbMATH Keywords: Steiner formula
MaRDI profile type: Publication
full work available at URL: s00208-003-0497-7
OpenAlex ID: W2005706098
DOI: 10.1007/S00208-003-0497-7
description: scientific article; zbMATH DE number 2078026