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.
@article{1176343801,
author = {Eddy, W. F. and Hartigan, J. A.},
title = {Uniform Convergence of the Empirical Distribution Function Over Convex Sets},
journal = {Ann. Statist.},
volume = {5},
number = {1},
year = {1977},
pages = { 370-374},
language = {en},
url = {http://dml.mathdoc.fr/item/1176343801}
}
Eddy, W. F.; Hartigan, J. A. Uniform Convergence of the Empirical Distribution Function Over Convex Sets. Ann. Statist., Tome 5 (1977) no. 1, pp. 370-374. http://gdmltest.u-ga.fr/item/1176343801/