An optimization problem for points on the sphere (master's dissertation)

Question

First, by means of a disclaimer, some background. I am entering the fourth and final year of an undergraduate master's degree in maths, and a quarter of the maximum credit for this year will be for a project on the subject of my choice, which I will describe below. In this project, I am allowed to cite and quote the work of others to any extent so long as it is correctly referenced. My current understanding is that final-year projects in pure maths tend to include little or none of the author's original research, but rather take the form of a summary with perhaps some relatively minor original investigation. I have checked that asking about my chosen subject on this website is within regulations, so long as I cite the thread, although I'm less clear on Math Overflow's attitude to this sort of thing. Assuming it's alright, please bear in mind that I might quote you unless you ask not to be quoted.

Moving on to the subject of the project, I am interested in "sphere-like" polyhedra. More specifically: given a polyhedron with a fixed number n of sides, I would like to minimize its surface area relative to its volume. Some preliminary research turned up Michael Goldberg's paper "The Isoperimetric Problem For Polyhedra", from Tohoku Mathematical Journal vol. 40 (1934), which can be found at http://staff.aist.go.jp/d.g.fedorov/Tohoku_Math_J_1934_40_226-236.pdf

This is the only paper in English that I was able to find directly addressing the problem. It cites a number of previous non-English papers, which it summarizes as not having established very much, the main result being that a solution polyhedron can be considered as a set of n points on the unit sphere, representing the centres of mass of the polyhedron's faces which are tangent to the sphere at this point. Then, if we define sphericity as inverse to the surface area of such a polyhedron circumscribed about a unit sphere, we are looking to maximize the sphericity given n.

Goldberg also makes some further progress on the problem, and conjectures that the solutions fall into a particular class of polyhedra, which he terms "medial". He published a later paper on the subject of medial polyhedra, which can be found at http://staff.aist.go.jp/d.g.fedorov/Tohoku_Math_J_1936_43_104-108.pdf

As an undergraduate making his first foray into mathematical research, I have little intuition for the directions I should be going in (and hence, apologies if any of the tags I've added are inappropriate); but I have considered looking at what happens if I add the additional constraint that the polyhedron should have a nontrivial symmetry group, as the few known results do (tetrahedron, triangular prism, cube, pentagonal prism, dodecahedron). To this end I thought using the stereographic projection, fixing one of the n points to be at infinity, might make it easier to study the symmetry and perhaps even view the points as roots of a polynomial. Then I may be able to provide new proofs for the known solutions, which might generalize so that I can investigate unknown solutions for small values of n, at least those which have nontrivial symmetry.

In the other direction, I am interested in the asymptotic behaviour of the solution set. Goldberg places some bounds on its behaviour in his first paper; I've been wondering about possible bounds on the minimum or maximum area of a single face in terms of n, or on the maximum number of edges it can have (I've got a gut feeling that most of the faces are going to be hexagons for large enough n). An important question is of course how sphericity behaves asymptotically with respect to n. While I expect the solution set will eventually become quite irregular, is it possible to find a well-behaved family of polyhedra for which sphericity behaves similarly, if not quite as well?

Finally, how if at all does this problem relate to other optimization problems for points on the sphere? The known solutions at least coincide, for example, with those for minimizing the energy of n point charges on a sphere (the Thomson problem), though I suspect the coincidence only occurs because n is small.

In summary, my question is this: are there any English-language publications, or translations, on this subject of which I am currently unaware? Has any further progress been made? If not, what avenues sound like they may be worth exploring? If anyone is interested in pursuing the problem themselves and would be willing to give me permission to cite their work I would of course be delighted.

Many thanks,

Robin

Answer 1

You might check out work by George Polya. In one of his "popular" books (I think Induction and analogy in mathematics) he defines the Isoperimetric Quotient (IQ) as volume squared over surface area cubed (perhaps times $36\pi$ to get a sphere to one). Amazingly, not all the platonic solids are optimum for their number of faces. That is at a simple level but a nice starting place. It looks like his book Isoperimetric inequalities in mathematical physics with Gábor Szegő addresses these things at a higher level. This is 60 years old but his ideas and exposition are famous. You could search for isoperimetric quotient (although you'll find stuff for plane curves) and also for references to Polya.

Answer 2

You might be interested in the recent article by Böröczky and Csikós concerning polytopes $P_n$ of $n$ facets with minimal surface area. They studied best approximations of a convex body $K\subset \mathbb E^d$ with $C^2$ boundary by circumscribed polytopes with respect to the surface area. Their main results can be roughly summarized as follows.

Theorem. Given a convex body $K$ in $\mathbb E^d$, $d \geq 2$, with $C^2$ boundary, let $P_n$ be a circumscribed polytope with $n$ facets that has minimal surface area. Denote by $\Xi_n$ a family of points where the $n$ facets of $P_n$ touch $\partial K$. Then

• $\{\Xi_n\}$ is uniformly distributed with respect to a density defined in terms of the the second fundamental form of $\partial K$;
• $$S(P_n) − S(K)\sim \frac{C_K}{n^{2/(d-1)}}\quad \mbox{as }\ n\to\infty,$$ where $S(K)$ ($S(P_n)$) is the surface area of $K$ ($S(P_n)$).

This work extends (and builds on) some of the earlier results by Fejes Tóth, Gruber, Ludwig and many others who studied optimal polytopal approximations of convex bodies in terms of volume and mean width. For a modern survey of these results and related references check out "Convex and Discrete Geometry" by Peter Gruber.