Let $\mathcal{P}$ be the convex hull of a point set $p_1, \dotsc, p_n$ (for simplicity, assume that no $p \in P$ lies in the convex hull of the other points.) Now, pick a point uniformly at random from the simplex $x_1 + \dotsc + x_n = 1, x_i \geq 0,$ and take $\phi=\sum x_i p_i.$ The point $\phi$ will be contained in $\mathcal{P}$ and in the simple case where $\mathcal{P}$ is a triangle, $\phi$ is uniformly distributed. In general, however, it is not: The two graphics are for the case $\mathcal{P}$ is a square. The first is a simple plot of a million points sampled from the distribution - the distribution looks like it is trying to be supported on the quadrangle spanned by the midpoints of the sides of the square.

The second is the density histogram:

which seems to indicate that the distribution has a bit of a peak at the center of gravity.

Any deep thoughts?