A dimension-free bound for isoperimetry of unconditional log-concave measures

Abstract

We prove a dimension-free Poincar\'e inequality for isotropic unconditional log-concave probability measures, establishing the Kannan-Lov\'asz-Simonovits (KLS) conjecture in the unconditional setting. The proof combines a new application of Dunkl operator theory combined with an appropriate transformation of the log-concave measure. Specifically, the proof follows by a direct spectral analysis of a new Dunkl-Langevin operator adapted to the transformed measure. The core ideas of the proof were generated by AI generative tools through interactions with the authors over several weeks. The authors refined and formalized the argument.