  Poincar{\'e} inequality is a fundamental property that rises naturally in
different branches of mathematics. The associated Poincar{\'e} constant plays a
central role in many applications, since it governs the convergence of various
practical algorithms. For instance, the convergence rate of the Langevin
dynamics is exactly given by the Poincar{\'e} constant. This paper investigates
a Riemannian version of Poincar{\'e} inequality where a positive definite
weighting matrix field (i.e. a Riemannian metric) is introduced to improve the
Poincar{\'e} constant, and therefore the performances of the associated
algorithm. Assuming the underlying measure is a \emph{moment measure}, we show
that an optimal metric exists and the resulting Poincar{\'e} constant is 1. We
demonstrate that such optimal metric is necessarily a \emph{Stein kernel},
offering a novel perspective on these complex but central mathematical objects
that are hard to obtain in practice. We further discuss how to numerically
obtain the optimal metric by deriving an implementable optimization algorithm.
The resulting method is illustrated on a few simple but nontrivial examples,
where solutions are revealed to be rather sophisticated. We also demonstrate
how to design efficient Langevin-based sampling schemes by utilizing the
precomputed optimal metric as a preconditioner.
