Given a point ${\bf x} = (x_1,x_2,\dots,x_n)$ in the orthosimplex $K = \{(x_1,x_2,\dots,x_n)\ : \ 0 \leq x_1 \leq x_2 \leq \dots \leq x_n \leq 1\}$, what proportion of a ball of radius $\epsilon$ around ${\bf x}$ lies in $K$, for small positive $\epsilon$? (This will be independent of $\epsilon$ once it's small enough.)
For instance, with $n=2$, $K$ is an isosceles right triangle. The proportion in question is 1/8 at two of the vertices of $K$, 1/4 at the remaining vertex of $K$, 1/2 for ${\bf x}$ lying on an edge of $K$, and 1 for ${\bf x}$ lying in the interior of $K$.