General \((p,q)\)-mixed projection bodies
type: Item
instance of: scholarly article
title: General \((p,q)\)-mixed projection bodies
zbMATH Open document ID: 1478.52002
author: Yibin Feng
author: Yanping Zhou
published in: Open Mathematics
publication date: 2021-11-29T00:00:00Z
review text: For \(K\in{\mathcal K}^n_o\) (the set of convex bodies in \({\mathbb R}^n\) containing the origin in the interior) and \(Q\in{\mathcal S}^n_o\) (the set of star bodies in \({\mathbb R}^n\)), and for \(p,q\in{\mathbb R}\), \textit{E. Lutwak} et al. [Adv. Math. 329, 85--132 (2018; Zbl 1388.52003)] have defined the \(L_p\) dual curvature measures \(\tilde C_{p,q}(K,Q,\cdot)\) by \[ \int_{S^{n-1}} g(v)\,\mathrm{d}\tilde C_{p,q}(K,Q,v) = \frac{1}{n} \int_{S^{n-1}} g(\alpha_K(u))h_K(\alpha_K(u))^{-p}\rho_K(u)^q\rho_Q(u)^{n-q}\,\mathrm{d}u\] for all continuous real functions \(g\) on the unit sphere \(S^{n-1}\), where \(\alpha,h,\rho\) are, respectively, the radial Gauss map, the support function, and the radial function. For \(p\ge 1\), \(q\in{\mathbb R}\), \(\tau\in[-1,1]\), the present authors define the general \((p,q)\)-mixed projection body \(\Pi_{p,q}^\tau(K,Q)\) by specifying its support function, \[ h\left (\Pi_{p,q}^\tau(K,Q),\cdot\right)^p = nc_{n,p}(\tau) \int_{S^{n-1}} \varphi_\tau(\langle \cdot,v\rangle)^p\,\mathrm{d}\tilde C_{p,q}(K,Q,v),\] where \(c_{n,p}(\tau)\) is an explicit constant, \(\varphi_\tau(t)=|t|+\tau t\), and \(\langle\cdot\,,\cdot\rangle\) is the scalar product. Special cases are the classical projection body and the \(L_p\) projection body. The paper finds out for which values of \(\tau\) the volume of \(\Pi_{p,q}^\tau(K,Q)\) or of its polar body attains its extreme values. It thus extends earlier results of \textit{C. Haberl} and \textit{F. E. Schuster} [J. Differ. Geom. 83, No. 1, 1--26 (2009; Zbl 1185.52005)].
reviewed by: Rolf Schneider
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 52A40
zbMATH DE Number: 7435407
zbMATH Keywords: projection body
zbMATH Keywords: \(L_p\) dual curvature measure
zbMATH Keywords: general \((p,q)\)-mixed projection body
zbMATH Keywords: volume inequalities
MaRDI profile type: Publication
full work available at URL: math-2020-0055
OpenAlex ID: W3089004981
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DOI: 10.1515/MATH-2020-0055

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