Extreme properties of quermassintegrals of convex bodies
type: Item
instance of: scholarly article
title: Extreme properties of quermassintegrals of convex bodies
zbMATH Open document ID: 1006.52003
DOI: 10.1007/BF02880133
published in: Science in China. Series A
publication date: 2002-08-15T00:00:00Z
review text: The authors prove inequalities involving quermassintegrals of projections of convex bodies. One main result says (after correcting a misprint) that, for unit vectors \(u_1,\dots,u_m\) in \(\mathbb{R}^n\) and real numbers \(c_1,\dots, c_m\) satisfying \(\sum^m_{i=1} c_iu_i\otimes u_i=I_n\) (identity), one has \[ W_j(K)^{n-j-1} \leq\omega_n^{-j/n} \prod^m_{i=1} \bigl\{W_j(K^{u_i}) \bigr \}^{c_i (n-j)/n} \] for every convex body \(K\) with interior points in \(\mathbb{R}^n\) and for \(0\leq j\leq n-1\). Here \(W_j\) is the \(j\)th quermassintegral and \(K^u\) denotes the image of \(K\) under orthogonal projection to \(u^\perp\) \((\omega_n\) is the volume of the \(n\)-dimensional unit ball). For \(j=0\), the result reduces to a theorem of \textit{K. Ball} [Trans. Am. Math. Soc. 327, No. 2, 891-901 (1991; Zbl 0746.52007)]. The proof follows the one by Ball, and in addition uses known inequalities for quermassintegrals. The paper contains further results on quermassintegrals of projections. Reviewer's remark. The other main result, Theorem 2 (also marred by misprints: the assumptions should be \(K\in\mathbb{K}^n\), \(C\in\mathbb{K}^n_s\); cf. Corollary 1), is contained in earlier results by \textit{G. D. Chakerian} and \textit{E. Lutwak} [Trans. Am. Math. Soc. 349, No. 5, 1811-1820 (1997; Zbl 0873.52010)].
reviewed by: Rolf Schneider
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 52A39
zbMATH DE Number: 1782129
zbMATH Keywords: Loomis-Whitney inequality
zbMATH Keywords: quermassintegrals
zbMATH Keywords: projections of convex bodies
author: Gangsong Leng
author: Liansheng Zhang
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02880133
OpenAlex ID: W2043545293
description: scientific article; zbMATH DE number 1782129