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Waists of balls in different spaces Akopyan, Arseniy
Description
Gromov and Memarian (2003--2011) have established the \emph{waist inequality} asserting that for any continuous map $f : \mathbb S^n \to \mathbb R^{n-k}$ there exists a fiber $f^{-1}(y)$ such that every its $t$-neighborhood has measure at least the measure of the $t$-neighborhood of an equatorial subsphere $\mathbb S^{k}\subset \mathbb S^n$. Going to the limit we may say that the $(n-k)$-volume of the fiber $f^{-1}(y)$ is at least that of the standard sphere $\mathbb S^{k}$. We extend this limit statement to the exact bounds for balls in spaces of constant curvature, tori, parallelepipeds, projective spaces and other metric spaces. By the volume of preimages for a non-regular map $f$ we mean its \emph{lower Minkowski content}, some new properties of which will be also presented in the talk. Joint work with Roman Karasev and Alfredo Hubard.
Item Metadata
Title |
Waists of balls in different spaces
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Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
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Date Issued |
2017-05-23T11:18
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Description |
Gromov and Memarian (2003--2011) have established the \emph{waist inequality} asserting that for any continuous map $f : \mathbb S^n \to \mathbb R^{n-k}$ there exists a fiber $f^{-1}(y)$ such that every its $t$-neighborhood has measure at least the measure of the $t$-neighborhood of an equatorial subsphere $\mathbb S^{k}\subset \mathbb S^n$.
Going to the limit we may say that the $(n-k)$-volume of the fiber $f^{-1}(y)$ is at least that of the standard sphere $\mathbb S^{k}$. We extend this limit statement to the exact bounds for balls in spaces of constant curvature, tori, parallelepipeds, projective spaces and other metric spaces.
By the volume of preimages for a non-regular map $f$ we mean its \emph{lower Minkowski content}, some new properties of which will be also presented in the talk.
Joint work with Roman Karasev and Alfredo Hubard.
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Extent |
31 minutes
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Subject | |
Type | |
File Format |
video/mp4
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Language |
eng
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Notes |
Author affiliation: IST Austria
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Series | |
Date Available |
2017-11-19
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0358005
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URI | |
Affiliation | |
Peer Review Status |
Unreviewed
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Scholarly Level |
Postdoctoral
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Rights URI | |
Aggregated Source Repository |
DSpace
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Rights
Attribution-NonCommercial-NoDerivatives 4.0 International