Absolute Continuity for Curvature Measures of Convex Sets I

✍ Scribed by Daniel Hug

• Publisher: John Wiley and Sons
• Year: 1998
• Tongue: English
• Weight: 923 KB
• Volume: 195
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19981950108

✦ Synopsis

Let us denote by C p ( K , .), r E {O,. . . , d -1)1 the curvature measures of a convex body K in the Euclidean space Ed with d 1. 2. According to Lebesgue's decomposition theorem the curvature measure of order r of K, C,(K, s), can be written as the sum of an absolutely continuous measure, C:(K, .), and a singular measure, C,d(K, . ) I with respect to (d-1)-dimensional Hausdorff measure. For example, if K is a polytope, then CF(K, .) = 0 and if K is sufficiently smooth, then C:(K, .) = 0. For a general convex body K , a description of C,O(K, .) in terms of geometric quantities is known. In the present paper, we provide a corresponding explicit representation for the singular part C,S(K, .). Further, denote by C ' ( K ) the set of r-singular boundary points of the convex body K. It is known that C'(K) has o-finite, but possibly infinite, r-dimensional Hausdorff measure. Provided that the singular part C,d(K, . ) of the curvature measure of order r vanishes, for a given convex body K, we prove that the r -dimensional Hausdorff measure of C ' ( K ) also vanishes. Examples show that in a certain sense this result is sharp. Analogous results are established for surface area measures and singular normal vectors.