Abstract
This paper presents connections between Gromov’s work on isoperimetry of waists and Milman’s work on the M-ellipsoid of a convex body. It is proven that any convex body \({K \subseteq \mathbb{R}^n}\) has a linear image \({\tilde{K}\subseteq \mathbb{R}^n}\) of volume one satisfying the following waist inequality: Any continuous map \({f:\tilde{K}\rightarrow \mathbb{R}^{\ell}}\) has a fiber \({f^{-1}(t)}\) whose \({(n-\ell)}\)-dimensional volume is at least \({c^{n-\ell}}\), where \({c > 0}\) is a universal constant. In the specific case where \({K = [0,1]^n}\) it is shown that one may take \({\tilde{K} = K}\) and \({c = 1}\), confirming a conjecture by Guth. We furthermore exhibit relations between waist inequalities and various geometric characteristics of the convex body K.
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Klartag, B. Convex geometry and waist inequalities. Geom. Funct. Anal. 27, 130–164 (2017). https://doi.org/10.1007/s00039-017-0397-8
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DOI: https://doi.org/10.1007/s00039-017-0397-8