Isoperimetry of waists and concentration of maps
type: Item
instance of: scholarly article
title: Isoperimetry of waists and concentration of maps
zbMATH Open document ID: 1044.46057
published in: Geometric and Functional Analysis. GAFA
publication date: 2003-08-27T00:00:00Z
full work available at URL: http://cds.cern.ch/record/623668
review text: Let \(S^n\) be the unit \(n\)-sphere. A measurable partition \(\Pi\) of \(S^n\) into convex subsets \(S=S_\pi,\;\pi\in \Pi\), is called convex if it equals the limit of consecutive refinements of finite partitions into convex subsets. Let \(\mu\) be a Borel probability measure on \(S^n\), \(\mathcal{M}_k^\prime\) the space of weak limits \(\mu^\prime\) of the restrictions \((\mu| U_i)/\mu(U_i)\) for all sequences of convex open \(U_i\)'s that Hausdorff converge to compact convex subsets \(S\subset S^n\) of codimension (exactly!) \(k\) and assume that \(S=\text{supp}\, \mu^\prime\). Let \(c^k_\bullet:\mathcal{M}_k^\prime\to S^n\) be a continuous mapping such that \(c^k_\bullet(\mu^\prime)\in S=\text{supp}\, \mu^\prime\) for all \(\mu^\prime\) (e.g., the center of mass of \(\mu^\prime\)). If \(\Pi\) is a convex partition of \(S^n\) into \(S_\pi\)'s with \(\text{codim}\, S_\pi\geq k\), then \(\mu\) induces the canonical system of probability measures \(\mu_\pi\) on almost all \(S_\pi\) and these \(\mu_\pi\) are convexly derived from \(\mu\) for almost all \(\pi\in \Pi\) [see \textit{A. Giannopoulos} and \textit{V. Milman}, Euclidean structure in finite dimensional spaces, in: Handbook of the Geometry of Banach Spaces, Vol. 1, 707--779 (2001; Zbl 1009.46004)], and thus -- via the ``center'' map \(c^k_\bullet\) -- the ``central'' subset \(C_\pi\) is well defined in \(S_\pi\) for almost all \(\pi\). If \(f:S^n\to \mathbb{R}^k\) is a continuous map -- under the above assumptions -- the author proves that there exist a value \(z\in \mathbb{R}^k\) and a convex partition \(\Pi\) of \(S^n\) into \(S_\pi\)'s of codimension \(\geq k\), such that the level \(Y_z:=f^{-1}(z)\subset S^n\) meets \(C_\pi\) for almost all \(\pi\in \Pi\). In particular, there exists a point \(z\in \mathbb{R}^k\) such that the spherical \(n\)-volumes of the \(\varepsilon\)-neighbours of the level \(Y_z\), denoted \(Y_z+\varepsilon\subset S^n\), satisfy \[ \text{Vol}(Y_z+\varepsilon)\geq \text{Vol}(S^{n-k…
Mathematics Subject Classification ID: 46T12
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 46M20
Mathematics Subject Classification ID: 52A55
zbMATH DE Number: 1971828
zbMATH Keywords: probability measure
zbMATH Keywords: Minkowski measure
zbMATH Keywords: waist
zbMATH Keywords: Gaussian space
zbMATH Keywords: log-concavity
zbMATH Keywords: fibration
zbMATH Keywords: convex partition
zbMATH Keywords: convex derivation of measures
zbMATH Keywords: (co)homolo\-gical partition
zbMATH Keywords: Borsuk-Ulam partition
zbMATH Keywords: filling
zbMATH Keywords: parametric partition
zbMATH Keywords: parametric waist
zbMATH Keywords: isoperimetric inequalities
zbMATH Keywords: iso-domination
reviewed by: Mircea Craioveanu
MaRDI profile type: Publication
OpenAlex ID: W2146453955
DOI: 10.1007/S000390300004
author: Misha Gromov
description: scientific article; zbMATH DE number 1971828