CHAPTER 1.8 - Convex Surfaces, Curvature and Surface Area Measures

Abstract

This chapter discusses convex surfaces, curvature, and surface area measures. The boundary of a convex body with interior points is a closed convex hypersurface. Because of the convexity assumption, many geometric properties of such a hypersurface can be proved without any additional analytic assumption. Even a kind of differential geometry without differentiability assumptions is possible for convex hypersurfaces. A convex body is called strictly convex if its boundary contains no segments, and smooth if through each boundary point there is only one supporting hyperplane. If a convex body is either not strictly convex or not smooth, its boundary points can be classified under different aspects, regarding the kind or degree of deviation from strict convexity or smoothness, respectively. Assertions connected to this deviation from strictly convex or smooth behavior are considered as describing first-order properties of convex hypersurfaces. Second-order properties are those referring to the curvature behavior. As no differentiability assumptions are made, principal curvatures in the sense of differential geometry in general cannot be defined. They exist, however, almost everywhere. Using curvature measures and their counterparts, the surface area measures, one can develop a curvature theory of convex bodies and obtain existence and uniqueness results in the spirit of classical differential geometry of convex hypersurfaces, but for general convex bodies.