**Problem instance:** A closed convex body $B\subset {\Bbb R}^n$ of volume 1; a point $p\in B$; and a real number $v\in(0,1)$.

**Objective:** Find the probability $P(B,v,p)$ that $p\in B'$, for $B'$ a randomly chosen closed convex subset of $B$ of volume $v$.

Of course to make this problem to make sense, I must define "choosing $B'$ at random."

I'll propose a definition or two myself. Still, I'd be interested in any solution to this problem relative to any reasonable definition.

So I propose taking $P(B,v,p)$ as the limit of probabilities $P_m(B,v,p)$ defined by restricting the range of $B'$. For $P_m(B,v,p)$, we pick $m$ points at random from $B$, get $B'$ by taking their convex hull, but condition on getting the correct volume $v$. By using Lebesgue measure on ${\Bbb R}^n$ to pick points at random from $B$, this all makes sense.

Alternatively (if it makes a difference?), we pick $m$ points at random from $B$, take their convex hull to get $B'$ and condition on getting volume $v$ and also on *having none of the $m$ points wind up in the interior of the hull.*

The sort of answer I'm looking for would single out the function $P(B,v,\cdot)$ in some reasonably nice way but not necessarily explicit way, as the solution of some differential or integral equation perhaps.