An inscribed polytope is a convex polytope whose vertices all lie on a common hypersphere. Fixing the hypersphere removes the freedom to enlarge the polytope by dilation. One extremal question asks for the largest content at a prescribed number of vertices.
For vertices on the unit hypersphere 
, Medgyes (2026) proved that the maximum
content with seven vertices
is 
.
No maximizer is a cyclic polytope. With eight
vertices, the maximum is 
, attained uniquely, up to orthogonal
transformations, by the regular 16-cell. The proof
uses certificates from sum-of-squares
optimization checked in exact arithmetic. Medgyes
(2026) reports AI assistance with mathematical discussions and implementation of
the computations.
