[math-fun] Failed(?) puzzle about convex objects
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)
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
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Andy Latto -
Warren D Smith