Discrete isoperimetric problems It is well-known that among all planar curves, the circle — invariant under $O(2)$ — has the best isoperimetric ratio. Similarly, among all $n$-gons, the regular $n$-gon — invariant under the dihedral group $D_n$ ($\lhd O(2)$) — has the best isoperimetric ratio. For the 2-D isoperimetric problem, it happens so that the solution of the discrete problem inherits as much symmetry as it deserves from that of the continuous problem, even when the discretization parameter $n$ is as small as 3.
Is anything known about the higher-dimensional cases? For example, if we optimize the isoperimetric ratio over all closed piecewise linear surfaces with the connectivity of any one of the five platonic solids, is the optimizer invariant under the corresponding symmetry group?
 A: You may find answers to some of your questions in the book "Regular Figures"       by Laszló Fejes Tóth, published in 1964. Following is the review of the book, as it appeared in the Mathematical Reviews.
(MR0165423
Fejes Tóth, L.
Regular Figures.
A Pergamon Press Book
The Macmillan Company, New York 1964 xi+339 pp.)
The appeal of symmetry, displayed in a regular polyhedron or a wall frieze pattern, is universal. The mathematics behind the symmetry, while having its roots in antiquity, is by no means exhausted. These are the two main impressions that one is left with after reading this work.
The book is divided into two parts. Part One, entitled “Systematology of the Regular Figures”, outlines the classical theory. This is, of course, a group-theoretic approach. The author begins by considering the discrete symmetry groups of the Euclidean plane. All seventeen plane crystallographic groups are illustrated. The three regular and eight semiregular (archimedean) tessellations are introduced. This leads naturally to circle-
packings and circle-coverings, which occupy much of the author’s attention later on. There follow chapters dealing with regular figures on the sphere and in the hyperbolic plane. The author then turns to Euclidean space of three dimensions, with a survey of regular and semiregular polyhedra, and the parallelohedra. Part One concludes with an introduction to regular polytopes of higher dimensions, especially those of dimension four.
Part Two, “Genetics of the Regular Figures”, aims at exhibiting the regular figures as solutions of various extremum problems. To cite the best-known example, the arrangement of circles in the densest close-packing of equal circles in the Euclidean plane is such that the centres are the vertices of the regular tessellation {3, 6}. This part of the book fairly bristles with tantalizing theorems, the statements of which are disarmingly simple, but whose proofs are often long and intricate. The author considers problems
of this nature in the Euclidean plane, on the sphere, in the hyperbolic plane, and in Euclidean spaces of three and higher dimensions, thus paralleling the material in Part One. There are, of course, many gaps in this approach; it is not nearly as complete as the classical theory given in Part One. But enough is given to make the approach convincing and attractive. It will surely wet the appetites of geometers for many years to
come.
Apart from the selection of the material itself, other noteworthy features of the book are the historical remarks and the many illustrations, which include twelve anaglyphs, complete with a viewer. There is a large bibliography and an adequate index.
F. A. Sherk
A: Let origin be the center of mass of the polyhedron $A.$ We know that a each $n-1$-dimensional facet contains a point $x_i$ such that the line through the origin is orthogonal to the facet. We know also that the polyhedron can be divided to cones with height $\|x_i\|_2.$ Let the $n-1$ dimensional measures of the facets be $|V_i|,$ corresponding to cone heights $\|x_i\|_2$ respectively. Volumes of the cones are then $\frac{|V_i|*\|x_i\|_2}{n}.$ And the volume of $$|A| = \sum_{i=1}^m \frac{|V_i|*\|x_i\|_2}{n}.$$ However, the surface measure is $$|\partial A| = \sum_{i=1}^m|V_i|.$$ So you can form $$\frac{|A|}{|\partial A|} = \frac{\sum_{i=1}^m |V_i|*\|x_i\|_2}{n\sum_{i=1}^m|V_i|}.$$ Set $|A|*n = 1.$ Then you can use the AGM-inequality to obtain that $$(\Pi_{i=1}^m |V_i|*\|x_i\|_2)^{1/m} \leq 1$$ with equality iff the cone sizes $(x_i*|V_i|)/n$ are all the same. (EDIT) It's clear that if you choose the average over $|V_i|$ then $\|x_i\|_2$ are all equal (for the maximum). But more importantly does this kind of polyhedron always exist? (EDIT)(EDIT) You can set $||x_i||_2 = 1 $ for $i = 1,\dots,m$ in order to notice that if those numbers are equal the ratio is always the same in a fixed dimension.
