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A lower bound on the waist of unit spheres of uniformly convex normed spaces

Part of: Normed linear spaces and Banach spaces; Banach lattices Spaces with richer structures

Published online by Cambridge University Press:  15 May 2012

Yashar Memarian
Yashar Memarian*
Affiliation:
Faculty of Mathematical and Physical Sciences, University College London, London, WC1E 6BT, UK (email: yashar.memarian@math.u-psud.fr)
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Abstract

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In this paper we give a lower bound on the waist of the unit sphere of a uniformly convex normed space by using the localization technique in codimension greater than one and a strong version of the Borsuk–Ulam theorem. The tools used in this paper follow ideas of Gromov in [Isoperimetry of waists and concentration of maps, Geom. Funct. Anal. 13 (2003), 178–215] and we also include an independent proof of our main theorem which does not rely on Gromov’s waist of the sphere. Our waist inequality in codimension one recovers a version of the Gromov–Milman inequality in [Generalisation of the spherical isoperimetric inequality to uniformly convex Banach spaces, Compositio Math. 62 (1987), 263–282].

Keywords

uniformly convexnormed spacewaistconvexly derived measuresisoperimetry

MSC classification

Secondary: 54E70: Probabilistic metric spaces 46B20: Geometry and structure of normed linear spaces
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 4 , July 2012 , pp. 1238 - 1264
Copyright
Copyright © Foundation Compositio Mathematica 2012

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