The functional form of the dual mixed volume
type: Item
instance of: scholarly article
title: The functional form of the dual mixed volume
zbMATH Open document ID: 1486.52018
author: Rigao He
author: Beifang Chen
author: Wei Wang
published in: Advances in Applied Mathematics
publication date: 2022-01-21T00:00:00Z
review text: The authors generalize the classical notion of dual mixed volume of star bodies to that of dual quasi-concave functions, which are defined as follows: \(f:\mathbb{R}^n\longrightarrow\mathbb{R}\) is called dual quasi-concave if \(f\bigl(\lambda x+(1-\lambda)y\bigr)\geq\min\bigl\{f(x),f(y)\bigr\}\) for \(\lambda\in(0,1)\) and \(x,y\in\mathbb{R}^n\) such that \(\langle x,y\rangle=\|x\|\,\|y\|\). Then, given \(n\) nonnegative upper semicontinuous dual quasi-concave functions \(f_i:\mathbb{R}^n\longrightarrow[0,M_i]\), \(i=1,\dots,n\), the dual mixed volume of \(f_1,\dots,f_n\) is defined as \[ \tilde{V}(f_1,\dots,f_n):=(M_1\cdots M_n)^{1/n}\int_0^1\tilde{V}\left(X_a\Bigl(\frac{f_1}{M_1}\Bigr),\dots,X_a\Bigl(\frac{f_n}{M_n}\Bigr)\right)da, \] where the functional \(\tilde{V}(\cdot,\dots,\cdot)\) in the integral is the classical dual mixed volume of the star bodies given by the superlevel sets of the functions \(f_i/M_i, i.e., \) \(X_a(f)=\{x\in\mathbb{R}^n:f(x)\geq a\}\). The main properties of this new functional are studied (monotonicity, linearity, continuity, polynomial expansion for the radial convolution of the functions...), and several classical inequalities are extended to this new setting, as the dual mixed Brunn-Minkowski inequality or the dual Aleksandrov Fenchel inequality.
reviewed by: Maria A. Hernández Cifre
Mathematics Subject Classification ID: 52A39
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 52A20
zbMATH DE Number: 7461718
zbMATH Keywords: radial Minkowski addition
zbMATH Keywords: quasi-concave function
zbMATH Keywords: dual mixed volume
zbMATH Keywords: dual Aleksandrov Fenchel inequality
zbMATH Keywords: dual quasi-concave
MaRDI profile type: Publication
full work available at URL: j.aam.2021.102305
OpenAlex ID: W4200419604
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The functional form of the dual mixed volume (DOI)
DOI: 10.1016/J.AAM.2021.102305