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:

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

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? }