  In this paper we develop tools for studying limit theorems by means of
convexity. We establish bounds for the discrepancy in total variation between
probability measures $\mu$ and $\nu$ such that $\nu$ is log-concave with
respect to $\mu$. We discuss a variety of applications, which include geometric
and binomial approximations to sums of random variables, and discrepancy
between Gamma distributions. As special cases we obtain a law of rare events
for intrinsic volumes, quantitative bounds on proximity to geometric for
infinitely divisible distributions, as well as binomial and Poisson
approximation for matroids.
