We present an efficient algorithm for uniformly sampling from an arbitrary compact body $\mathcal{X} \subset \mathbb{R}^n$ under isoperimetry and a natural volume growth condition. This result significantly generalizes known results for convex bodies and star-shaped bodies. The complexity of the algorithm is polynomial in the dimension, the Poincaré constant of the uniform distribution on $\mathcal{X}$, and the volume growth constant of the set $\mathcal{X}$.
Blogger's Review: This algorithm provides a fresh perspective on nonconvex sampling problems, especially in high-dimensional spaces, effectively addressing the sampling needs of complex shapes. Its polynomial complexity guarantee also enhances its feasibility in practical applications, showcasing the significant role of geometry in computer science.