Generalized curvature measures and singularities of sets with positive reach

Abstract

For a convex body K in Euclidean space ℝ^d (d≥2) and for r ∈ {0,…, d – 1}, let Σ^r(K) be the set of r-singular boundary points of K. It is known that Σ^r(K) is countably r-rectifiable and hence has σ-finite r-dimensional Hausdorff measure. We obtain a quantitative improvement of this result, taking into account the strength of the singularities. Denoting by Σ^r(K, τ) the set of those r-singular boundary points of K at which the spherical image has (d–1– r)-dimensional Hausdorff measure at least τ>0, we establish a finite upper bound for the r-dimensional Hausdorff measure of Σ^r(K, τ). This estimate is deduced from an identity that connects Hausdorff measures of spherical images of singularities to the generalized curvature measure of the convex body K. The latter relation is, in fact, proved for the class of sets with positive reach. For convex bodies, similar result as for singular boundary points are obtained for singular normal vectors. We also consider the disintegration of generalized curvature measures with respect to projections onto the components of the product space ℝ^d × S^{d – 1}.