  For a $d$-dimensional log-concave distribution $\pi(\theta) \propto
e^{-f(\theta)}$ constrained to a convex body $K$, the problem of outputting
samples from a distribution $\nu$ which is $\varepsilon$-close in
infinity-distance $\sup_{\theta \in K} |\log \frac{\nu(\theta)}{\pi(\theta)}|$
to $\pi$ arises in differentially private optimization. While sampling within
total-variation distance $\varepsilon$ of $\pi$ can be done by algorithms whose
runtime depends polylogarithmically on $\frac{1}{\varepsilon}$, prior
algorithms for sampling in $\varepsilon$ infinity distance have runtime bounds
that depend polynomially on $\frac{1}{\varepsilon}$. We bridge this gap by
presenting an algorithm that outputs a point $\varepsilon$-close to $\pi$ in
infinity distance that requires at most $\mathrm{poly}(\log
\frac{1}{\varepsilon}, d)$ calls to a membership oracle for $K$ and evaluation
oracle for $f$, when $f$ is Lipschitz. Our approach departs from prior works
that construct Markov chains on a $\frac{1}{\varepsilon^2}$-discretization of
$K$ to achieve a sample with $\varepsilon$ infinity-distance error, and present
a method to directly convert continuous samples from $K$ with total-variation
bounds to samples with infinity bounds. This approach also allows us to obtain
an improvement on the dimension $d$ in the running time for the problem of
sampling from a log-concave distribution on polytopes $K$ with infinity
distance $\varepsilon$, by plugging in TV-distance running time bounds for the
Dikin Walk Markov chain.
