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November 2018 The KLS isoperimetric conjecture for generalized Orlicz balls
Alexander V. Kolesnikov, Emanuel Milman
Ann. Probab. 46(6): 3578-3615 (November 2018). DOI: 10.1214/18-AOP1257


What is the optimal way to cut a convex bounded domain $K$ in Euclidean space $(\mathbb{R}^{n},|\cdot|)$ into two halves of equal volume, so that the interface between the two halves has least surface area? A conjecture of Kannan, Lovász and Simonovits asserts that, if one does not mind gaining a universal numerical factor (independent of $n$) in the surface area, one might as well dissect $K$ using a hyperplane. This conjectured essential equivalence between the former nonlinear isoperimetric inequality and its latter linear relaxation, has been shown over the last two decades to be of fundamental importance to the understanding of volume-concentration and spectral properties of convex domains. In this work, we address the conjecture for the subclass of generalized Orlicz balls

\[K=\{x\in\mathbb{R}^{n};\sum_{i=1}^{n}V_{i}(x_{i})\leq E\},\] confirming its validity for certain levels $E\in\mathbb{R}$ under a mild technical assumption on the growth of the convex functions $V_{i}$ at infinity [without which we confirm the conjecture up to a $\log(1+n)$ factor]. In sharp contrast to previous approaches for tackling the KLS conjecture, we emphasize that no symmetry is required from $K$. This significantly enlarges the subclass of convex bodies for which the conjecture is confirmed.


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Alexander V. Kolesnikov. Emanuel Milman. "The KLS isoperimetric conjecture for generalized Orlicz balls." Ann. Probab. 46 (6) 3578 - 3615, November 2018.


Received: 1 December 2016; Revised: 1 January 2018; Published: November 2018
First available in Project Euclid: 25 September 2018

zbMATH: 06975494
MathSciNet: MR3857863
Digital Object Identifier: 10.1214/18-AOP1257

Primary: 46B07, 52A23, 60D05

Rights: Copyright © 2018 Institute of Mathematical Statistics


Vol.46 • No. 6 • November 2018
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