Absolute Continuity for Curvature Measures of Convex Sets I

Abstract

AbstractLet us denote by Cr(K, ·), r ∈ {0,…, d ‐ 1}, the curvature measures of a convex body K in the Euclidean space ℝd with d ≥ 2. According to Lebesgue's decomposition theorem the curvature measure of order r of K, Cr(K, ·), can be written as the sum of an absolutely continuous measure, Car(K, ·), and a singular measure, Csr(K, ·), with respect to (d ‐ 1) ‐ dimensional Hausdorff measure. For example, if K is a polytope, then Car(K, ·) = 0 and if K is sufficiently smooth, then Csr(K, ·) = 0. For a general convex body K, a description of Car(K, ·) in terms of geometric quantities is known. In the present paper, we provide a corresponding explicit representation for the singular part Csr(K, .). Further, denote by Σr (K) the set of r‐singular boundary points of the convex body K· It is known that Σr(K) has σ‐finite, but possibly infinite, r ‐ dimensional Hausdorff measure. Provided that the singular part Csr(K, ·) of the curvature measure of order r vanishes, for a given convex body K, we prove that the r ‐ dimensional Hausdorff measure of Σr (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.