Can anyone think of a multivariate (i.e. non--univariate) cumulative distribution function $F$ that is _concave_ on every convex subset where $F$ is positive? In the univariate case (i.e. d=1,) and on the positive real line, this corresponds to the class of all non--increasing desnities on the positive real line.
In higher dimensions (i.e. let's say d=2) does anyone know an example of a concave CDF or if they know if such a CDF does not exist?
[Concavity in higher dimensions is the same as in the one dimension; i.e. for vectors $x$ and $y$ and a $a \in (0,1)$ we should have $F(ax + (1-a)y)\leq aF(X) + (1-a)F(y)$.]
Note: for non--LaTeX users "\leq" means "<=". Thanks!