Let C be a compact convex subset of 3-dimensional Euclidean space E(3) whose interior is non-empty and whose diameter is d. What is the largest volume that C can have if every subset of C that is a straight line segment of length d is a subset of the boundary of C (or-equivalently-if no chord of C that contains an interior point of C has length d)? It looks as though this maximum might be close to the volume of a right circular cone whose base radius is d/2 and whose height is ((3^(1/2))/2)*d. I am unable to say more. Perhaps there is no maximum volume-only a least upper bound. Although C is a convex body, it is not what would be called stricly convex. To keep things simple I am avoiding any discussion of higher dimensional versions of this question.

  • $\begingroup$ The equivalent form of the question is to maximize the volume of a convex set of diameter $d$ such that it contains <i>one</i> segment of length $d$ on its surface. (Then you may compress everything a bit to make this segment unique.) $\endgroup$ – Ilya Bogdanov Sep 1 '13 at 10:28

I would suggest yet another larger set. Take two points $A$ and $B$ in the plane with $|AB|=d$. Let $S$ be the set of all points $X$ in this plane lying in one halfplane defined by $AB$ and satisfying $|XA|,|XB|\leq d$. Then rotate $S$ around the perpendicular bisector of $AB$; you get the convex body containing your cone. Then, after smashing it a bit you get a desired set --- still of larger volume.


It seems you could curve the base of the cone, making it a sector of a sphere of radius $r > d$ slightly larger than $d$, centered at a point $c$ directly above the apex of the cone:
(Arrows indicate segments of length $d$ that cannot fit.) If this is correct, the max volume can be approached but not achieved, as you suggest.

Update. And here is Ilya Bogdanov's larger body, a half-lune:

  • $\begingroup$ Thanks to all of you for your many interesting examples. They all reinforce my feeling there is a least upper bound $\endgroup$ – Garabed Gulbenkian Sep 2 '13 at 17:57
  • $\begingroup$ Thanks to all of you for your many interesting examples. They all reinforce my feeling that there is a least upper bound on the volume but no actual maximumum volume. But apart from playing around with examples, I can see no way to make further progress on this problem and actually prove something. It is interesting that a solid hemisphere having diameter d is a convex set of this kind which is larger than the right circular cone I mentioned. But one cannot seem to add volume to the hemisphere as one can to the cone. $\endgroup$ – Garabed Gulbenkian Sep 2 '13 at 18:22

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.