Re: [math-fun] Failed(?) puzzle about convex objects

On Wed, Dec 17, 2014 at 4:27 PM, Warren D Smith <warren.wds@gmail.com> wrote:

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.

It is? Take B to be the interval [1,2], while A(x) is the interval [-x/2, x/2]. This is not convex.

So... doesn't seem like there is any nice theorem of this ilk.

How about if we define f(x) to be the max, over all translations of A(x), of the volume of A(x) intersect B. Does that work? Andy