New geometric inequalities for hedgehogs
type: Item
instance of: scholarly article
title: New geometric inequalities for hedgehogs
zbMATH Open document ID: 0955.52003
author: Yves Martinez-Maure
published in: Archiv der Mathematik
publication date: 2001-03-04T00:00:00Z
review text: A hypersurface \(H_h\) in Euclidean \((n+1)\)-space \(\mathbb{E}^{n+1}\) is called a hedgehog if it is the envelope of hyperplanes \(\langle x,p\rangle= h(p)\) where \(h\) is a function in \(\mathbb{C}^2(S^n,\mathbb{R})\), \(S^n\) the unit sphere in \(\mathbb{E}^{n+1}\). This is a generalization of the case where \(h\) is the support function of a convex body \(K\) and \(H_h\) coincides with the surface of \(K\). The author derives the Alexandrov-Fenchel inequality \[ \nu(K_1, \dots, K_{n +1}) \geq\nu (K_1,K_1, K_3,\dots, K_{n+1})\cdot \nu(K_2,K_2,K_3, \dots, K_{n+1}), \] where \(K_3,\dots, K_{n+1}\) denote ordinary convex bodies, only \(K_1\) and \(K_2\) are hedgehogs and \(\nu(\cdots)\) is a generalization of the mixed volume of convex bodies. Further inequalities for hedgehogs are derived from the one above. In particular, in \(\mathbb{E}^3\) a so far unknown inequality for a convex body \(K\) of the type \(V\geq f(S,M)\) is established, under a certain assumption on a hedgehog associated to the support function of \(K\), where \(V,S,M\) denote volume, surface area, integral of the mean curvature of \(K\), respectively. Brecause of the additional assumption, the conjecture of \textit{J. R. Sangwine-Yager} [Am. Math. Mon. 96, No. 3, 223-237 (1989; Zbl 0669.52008)] concerning the boundary of the Blaschke diagram for \(K\) is still open.
reviewed by: Bernd Kind
Mathematics Subject Classification ID: 52A39
Mathematics Subject Classification ID: 52A40
zbMATH DE Number: 1333333
zbMATH Keywords: hedgehog
zbMATH Keywords: Alexandrov-Fenchel inequality
zbMATH Keywords: Blaschke diagram
MaRDI profile type: Publication
full work available at URL: s000130050354
OpenAlex ID: W2024380224
DOI: 10.1007/S000130050354
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description: scientific article; zbMATH DE number 1333333