Polytope Sampling at the Speed of Isoperimetry

We give faster algorithms for sampling from the uniform distribution over a polytope. Our main result is an algorithm that, when applied to a polytope with $m$ constraints in $n$ dimensions, uses $m\cdot \mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}\log (\frac{m}{\delta })$ iterations, each running in time $O(m{n}^{\omega -1})$, to return a sample within total variation distance $\delta$ to the uniform distribution. When $m=O(n)$ and $\delta =\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}(\frac{1}{n})$, our runtime improves upon all prior works by an $\mathrm{\Omega }(n^{2\mathrm{/}3})$ factor. Our sampler is based on the recent \emph{functional stochastic localization} framework of [GST26], and achieves an iteration count nearly matching the Poincaré constant of the uniform distribution with respect to the logarithmic barrier.