Refined convexity and special cases of the Blaschke-Santalo inequality
type: Item
instance of: scholarly article
title: Refined convexity and special cases of the Blaschke-Santalo inequality
zbMATH Open document ID: 1017.52002
author: David Borwein
author: Jonathan M. Borwein
author: Roland Girgensohn
author: G. Fee
publication date: 2003-08-19T00:00:00Z
Mathematics Subject Classification ID: 52A22
Mathematics Subject Classification ID: 26B25
Mathematics Subject Classification ID: 33B15
Mathematics Subject Classification ID: 52A38
zbMATH DE Number: 1703258
zbMATH Keywords: Psi function
zbMATH Keywords: Gamma function
zbMATH Keywords: modified convexity
zbMATH Keywords: volume of \(p\)-balls
zbMATH Keywords: Blaschke-Santalo inequality
zbMATH Keywords: substitution norm
MaRDI profile type: Publication
full work available at URL: mia-04-56
OpenAlex ID: W2014349068
DOI: 10.7153/MIA-04-56
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published in: Mathematical Inequalities \& Applications
review text: The authors use the following modified version of convexity of functions: If \(\varphi\) is a real, continuous strictly monotonic function defined on an interval \(I\), then \(M:I^2 \rightarrow I\) defined as NEWLINE\[NEWLINE M(x,y):=\varphi^{-1}((\varphi(x) + \varphi(y))/2) NEWLINE\]NEWLINE is called a quasi-arithmetic mean. NEWLINENEWLINENEWLINESimilarly NEWLINE\[NEWLINE M^{(\lambda)}(x,y):= \varphi^{-1}(\lambda \varphi (x) + (1-\lambda)\varphi (y)) NEWLINE\]NEWLINE for \(\lambda \in [0,1]\) is the weighted version of \(M\). NEWLINENEWLINENEWLINEFor any two quasi-arithmetic means \(M\), \(N\), a function \(f:I \rightarrow J\) is called \((M,N)\)-convex if it satisfies NEWLINE\[NEWLINE f(M^{\lambda}(x,y)) \leq N^{\lambda}(f(x),f(y)) NEWLINE\]NEWLINE for all \(x,y \in I\) and for all \(\lambda \in [0,1]\). NEWLINENEWLINENEWLINEUsing this modified notion of convexity the authors derive the \(\|. \|_p\)-norm version of the classical Blaschke-Santalo inequality for polar volumes. They also give analogs for the \((p,q)\)-substitution norm.
description: scientific article; zbMATH DE number 1703258