Hilbert volume in metric spaces. I
type: Item
instance of: scholarly article
title: Hilbert volume in metric spaces. I
zbMATH Open document ID: 1243.53077
published in: Central European Journal of Mathematics
publication date: 2012-05-31T00:00:00Z
full work available at URL: https://arxiv.org/abs/1811.04332
review text: Let \(Df\) denote the Jacobian matrix of a differentiable map \(f: \mathbb R^n\to\mathbb R^n\). One way to quantify the infinitesimal dilation of \(f\) is to consider the operator norm \(\|Df\|\). This corresponds to the local form of the Lipschitz constant \(\text{Lip}\,f=\sup_{a,b} \frac{|f(a)-f(b)|}{|a-b|}\), and thus makes sense in general metric spaces. Another natural, and often more convenient way to measure dilation is the Hilbert-Schmidt norm \(\|Df\|_{HS}\). However, the latter does not immediately generalize to metric spaces. The present article developes such a generalization and uses it to derive several previous results in a unified and elegant way. Instead of trying to describe the construction in full generality, let us consider the special case of Lipschitz maps \(f: X \to {\mathbb{R}}^n\), where \(X\) is a metric space. Let \(\mu\) be a measure on the set \(\mathcal P\) of all rank-\(1\) projections, which can be identified on the \((n-1)\)-dimensional projective space. Of particular importance are the measures \(\mu\) for which \(\int_{\mathcal P}p(x)\,d\mu = x\) for all \(x\in \mathbb R^n\). Such a measure is called an \textit{axial partition of unity}; a related term in harmonic analysis is a \textit{Parseval frame}. The \(L_2\)-dilation of \(f\) with respect to \(\mu\) is \[ \|\text{dil}^* f\|_{L_2(\mu)}=\left(\int_{\mathcal P} \text{Lip}^2(p\circ f) \,d\mu(p)\right)^{1/2}. \] In terms of frames, this definition means that one applies the analysis operator to \(f\) and measures the Lipschitz constant of the output. Another approach proceeds via the synthesis operator: One can consider all Lipschitz maps \(\tilde f: X\to L_2(\mathcal P,\mu)\) from which \(f\) can be synthesized and defines \[ \left\|\widetilde{\text{dil}}^* f\right\|_{L_2(\mu)} = \inf_{\tilde f} \left(\int_{\mathcal P} \text{Lip}^2(\tilde f(\cdot,p))\,d\mu(p) \right)^{1/2}. \] For every axial partition of unity one has \(\|\widetilde{\text{dil}}^* f\|_{L_2(\mu)} \leq \|\text{dil…
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 53C22
Mathematics Subject Classification ID: 53C24
zbMATH DE Number: 6039956
zbMATH Keywords: Riemannian volume
zbMATH Keywords: Lipschitz maps
zbMATH Keywords: Besicovitch inequality
zbMATH Keywords: John's ellipsoid
reviewed by: Leonid V. Kovalev
MaRDI profile type: Publication
OpenAlex ID: W2044707807
arXiv ID: 1811.04332
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DOI: 10.2478/S11533-011-0143-7
author: Misha Gromov
description: scientific article; zbMATH DE number 6039956