KLS Conjecture for Isoperimetric Constant of Isotropic Convex Bodies

Establish that for every convex body K in isotropic position (i.e., the uniform distribution over K has identity covariance), the Cheeger/isoperimetric constant C_K of the uniform distribution on K is bounded by an absolute O(1) constant independent of dimension.

Background

The paper discusses conductance-based analyses of Markov chains for sampling from convex bodies, where bounds depend on isoperimetric quantities of the uniform distribution over the body. An archetypal inequality involves an isoperimetric constant C_K. The KLS (Kannan–Lovász–Simonovits) conjecture asserts a dimension-free bound on this constant for isotropic convex bodies.

A resolution of the KLS conjecture would sharpen Poincaré and related functional inequalities for uniform measures on convex bodies, directly improving mixing-time guarantees for geometric random walks and diffusion-based samplers analyzed through isoperimetry, including the algorithm proposed in this paper.

References

The KLS conjecture posits that C_{\mc K} = O(1) for any convex body \mc K in isotropic position, i.e., under the normalization that a random point from \mc K has identity covariance.

— In-and-Out: Algorithmic Diffusion for Sampling Convex Bodies  (2405.01425 - Kook et al., 2024) in Section 1 (Introduction)