The empirical distribution function $P_n$ converges with probability 1 to a true distribution $P$ in $R^k$, uniformly over measurable convex sets, if and only if $P$ is a countable mixture of distributions, each of which is carried by a flat and gives zero probability to the relative boundaries of convex sets included in the flat.
"Uniform Convergence of the Empirical Distribution Function Over Convex Sets." Ann. Statist. 5 (2) 370 - 374, March, 1977. https://doi.org/10.1214/aos/1176343801