  Uniform sampling over a convex body is a fundamental algorithmic problem, yet
the convergence in KL or R\'enyi divergence of most samplers remains poorly
understood. In this work, we propose a constrained proximal sampler, a
principled and simple algorithm that possesses elegant convergence guarantees.
Leveraging the uniform ergodicity of this sampler, we show that it converges in
the R\'enyi-infinity divergence ($\mathcal R_\infty$) with no query complexity
overhead when starting from a warm start. This is the strongest of commonly
considered performance metrics, implying rates in $\{\mathcal R_q,
\mathsf{KL}\}$ convergence as special cases.
  By applying this sampler within an annealing scheme, we propose an algorithm
which can approximately sample $\varepsilon$-close to the uniform distribution
on convex bodies in $\mathcal R_\infty$-divergence with
$\widetilde{\mathcal{O}}(d^3\, \text{polylog} \frac{1}{\varepsilon})$ query
complexity. This improves on all prior results in $\{\mathcal R_q,
\mathsf{KL}\}$-divergences, without resorting to any algorithmic modifications
or post-processing of the sample. It also matches the prior best known
complexity in total variation distance.
