In N-dimensional space, we have a convex object A with unit volume. We scale it by x^(1/N) so it now has volume=x, and intersect it with convex object B. QUESTION: Is the volume of the resulting object, a convex (i.e. concave-down) function of x? When N=1, answer obviously is yes. Also, answer is yes for any N>=1 if A and B are suitable (same-oriented) hypercubes. Also if B is a halfspace. Or whole space. But answer is "no" for each N>=2 if A and B are suitable (different-oriented) hypercubes. So... that wasn't the right question. REVISED QUESTION: same thing, but demand both A and B contain the origin 0 inside. Nope, a counterexample is A=circle, B=square, N=2. So... doesn't seem like there is any nice theorem of this ilk. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)