Abstract: This chapter focuses on convexity and differential geometry. Alexandrov–Fenchel–Jessen inequalities hold for arbitrary compact convex bodies. The chapter presents differential geometric proof of the Alexandrov–Fenchel–Jessen inequalities. It explains the way by which the notion of principal curvatures or principal radii of curvature of a convex hypersurface F of class C 2 in d can be extended to the case of an arbitrary convex hypersurface. The importance of convexity in differential geometry consists in the fact that some differentiability assumptions can be removed if a differentiable hypersurface F of d is convex—that is, the point set x ( M ) lies in the boundary of some suitable closed convex set K of d . This way, convex geometric generalizations of notions and theorems in differential geometry are obtained. Conversely, if the boundary of a closed convex set K of d is the point set of some differentiable hypersurface F , it is not hard to compute convex geometric entities and to prove convex geometric theorems in a differential geometric manner.
Publication Year: 1993
Publication Date: 1993-01-01
Language: en
Type: book-chapter
Indexed In: ['crossref']
Access and Citation
Cited By Count: 24
AI Researcher Chatbot
Get quick answers to your questions about the article from our AI researcher chatbot