Total positive curvature of hypersurfaces with convex boundary

Abstract

We prove that if Σ is a compact hypersurface in Euclidean space Rⁿ, its boundary lies on the boundary of a convex body C, and meets C orthogonally from the outside, then the total positive curvature of Σ is bigger than or equal to half the area of the sphere S^n-1. Also, we obtain necessary and sufficient conditions for the equality to hold.