The Annals of Probability

The KLS isoperimetric conjecture for generalized Orlicz balls

Alexander V. Kolesnikov and Emanuel Milman

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Abstract

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.

Article information

Source
Ann. Probab., Volume 46, Number 6 (2018), 3578-3615.

Dates
Received: December 2016
Revised: January 2018
First available in Project Euclid: 25 September 2018

Permanent link to this document
https://projecteuclid.org/euclid.aop/1537862441

Digital Object Identifier
doi:10.1214/18-AOP1257

Mathematical Reviews number (MathSciNet)
MR3857863

Zentralblatt MATH identifier
06975494

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 52A23: Asymptotic theory of convex bodies [See also 46B06] 46B07: Local theory of Banach spaces

Keywords
KLS conjecture spectral-gap convex bodies generalized Orlicz balls

Citation

Kolesnikov, Alexander V.; Milman, Emanuel. The KLS isoperimetric conjecture for generalized Orlicz balls. Ann. Probab. 46 (2018), no. 6, 3578--3615. doi:10.1214/18-AOP1257. https://projecteuclid.org/euclid.aop/1537862441


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