# Does this formula for caliper diameter hold for concave polyhedra?

I recently asked on MathOverflow and also asked several people I know to prove the following:

How do I prove that the average caliper diameter of the polyhedron across all possible rotations is given by this formula: $$\sum_{e\in E} L_e(\pi - \delta_e)/(4\pi)$$

(see here for more information). Several people gave me proofs for which I'm grateful but there seemed to be some conflict on whether or not this equation is true for both convex and concave polyhedra.

I thought I'd start a new question for this as I don't want to confuse the purpose of the old thread... is anyone able to tell me is this equation true for concave polyhedra?/provide a counterexample if not?

No, this equation is false for non-convex polyhedra. Take a cube and remove from inside of it a smaller cube. The resulting body has the same mean width (caliper diameter), but the sum of angles times the edges is different (there are negative summands coming).

If one wants a counterexample where the body is homeomorphic to a ball, then one can choose the inner cube to be removed very close to the boundary of the outer cube and make a short and narrow tunnel from the cavity to the outside. The tunnel contribution to the sum is small, so the sum is still different from that for the solid cube.

The mean width can be extended to unions of convex bodies by a sort of inclusion-exclusion formula. For this, see for example Section 5 of

Klain, Daniel A.; Rota, Gian-Carlo, Introduction to geometric probability, Lezioni Lincee. Cambridge: Cambridge University Press. Rome: Accademia Nazionale dei Lincei, xiv, 178 p. (1997). ZBL0896.60004.

• Hmmm yes thank you! Do you have any notion of what the formula I have posted could represent for a concave shape if not the mean caliper diameter? – JDoe2 Aug 6 '18 at 15:39

The mean caliper formula for both convex and concave polyhedra is considered in [1], page 513. For a concave edge the angle $\pi-\delta_e$ is negative.

[1] Thinh Le and Qiang Du, A generalization of the three-dimensional MacPherson-Srolovitz formula, Comm. Math. Sci. 7, 511 (2009).

• A counterexample someone suggested was an object like this: 5.imimg.com/data5/NQ/CK/MY-30031960/… Surely this should have the same calliper dimension to the same box but filled, but this equation I think gives different answers... I'm not sure - am I applying it incorrectly? – JDoe2 Jul 19 '18 at 21:11