# Cyclic polygons generalized to higher dimensions

Many theorems hold for cyclic polygons—convex polygons inscribed in a circle. Perhaps the most basic is this, from the reference cited below:

Theorem. There exists a cyclic polygon of $n \ge 3$ sides of lengths $\ell_i > 0$ if and only if each $\ell_k$ is less than the sum of the other lengths. And this polygon is unique.

(Wikipedia image from article: Circumscribed circle.)

Kouřimská, Hana, Lara Skuppin, and Boris Springborn. "A variational principle for cyclic polygons with prescribed edge lengths." Advances in Discrete Differential Geometry. Springer Berlin Heidelberg, 2016. 177-195.

My question is:

Q. What is the closest higher-dimensional analog of this theorem? E.g., in $\mathbb{R}^3$ the areas would be prescribed.

I am familiar with Minkowski's theorem on the existence of a polytope realizing given facet areas/volumes and facet normals. What I am wondering is: If one assumes the polytope is inscribed in a sphere, can we reduce the information needed to justify existence/uniqueness? In other words, can Minkowski's theorem be "strengthened" by presuming the inscribed-in-a-sphere condition?

Related: Japanese Theorem” on cyclic polygons: Higher-dimensional generalizations?

## 1 Answer

The solution to Minkowski's problem already produces a polytope circumscribed around a sphere. Insisting that it be also inscribed may be asking for a little too much. However, if you are happy with replacing "inscribed" by "circumscribed", Minkowski's theorem is your man.

For inscribed polyhedra there are other sorts of results, in particular my results on the dihedral angles of ideal polyhedra in $\mathbb{H}^3,$ so if you use those instead of areas, you are golden. In $\mathbb{E}^3$ there is less information.

Rivin, Igor, A characterization of ideal polyhedra in hyperbolic 3-space, Ann. Math. (2) 143, No.1, 51-70 (1996). ZBL0874.52006.

• "Insisting that it be also inscribed may be asking for a little too much." Yes, you may be right. Thanks for the reference to your ideal polyhedra paper. – Joseph O'Rourke Sep 3 '17 at 23:49