Sufficiency of Poincaré inequality and bounded outer isoperimetry for efficient sampling

Establish that for any compact set X ⊂ R^n whose uniform distribution π ∝ 1_X satisfies a Poincaré inequality and whose outer isoperimetry ξ(X) ≡ area(∂X)/Vol(X) is bounded, there exists a polynomial-time algorithm for generating samples from π.

Background

The paper proves that the In-and-Out algorithm can sample efficiently from a broad class of nonconvex bodies under two assumptions: the uniform distribution on X satisfies a Poincaré inequality and X satisfies an (α,β)-volume growth condition. This strictly generalizes prior convex and star-shaped cases.

The authors speculate that the volume growth condition might be further weakened to a purely geometric bound on the outer isoperimetry ξ(X)=area(∂X)/Vol(X) while retaining efficient sampling guarantees, but this remains unproven.