Hadamard’s Characterization of the Ovaloids

Abstract

If p and q are points in Eⁿ, then $\overline {pq}$ denotes the line segment between p and q. A set S ⊂ Eⁿ is convex if for every p ∈ S and q ∈ S, $\overline {pq}$ ⊂S. A convex body is a compact convex set with a non-empty interior. It is easy to show that a convex body is homeomorphic to a solid sphere (but we will not need this fact). In these notes we will assume in addition that the boundary surface of a convex body in E³ is several times differentiable.