# 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?

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.