Year: 2000
Type: article
Abstract: For a given convex (semi-convex) function u u , defined on a nonempty open convex set Ω ⊂ R n \Omega \subset \mathbf {R}^n , we establish a local Steiner type formula, the coefficients of which are nonnegative (signed) Borel measures. We also determine explicit integral representations for these coefficient measures, which are similar to the integral representations for the curvature measures of convex bodies (and, more generally, of sets with positive reach). We prove that, for r ∈ { 0 , … , n } r\in \{0,\ldots ,n\} , the r r -th coefficient measure of the local Steiner formula for u u , restricted to the set of r r -singular points of u u , is absolutely continuous with respect to the r r -dimensional Hausdorff measure, and that its density is the ( n − r ) (n-r) -dimensional Hausdorff measure of the subgradient of u u . As an application, under the assumptions that u u is convex and Lipschitz, and Ω \Omega is bounded, we get sharp estimates for certain weighted Hausdorff measures of the sets of r r -singular points of u u . Such estimates depend on the Lipschitz constant of u u and on the querma
Authors: Andrea Colesanti, Daniel Hug
Institutions: University of Florence, University of Freiburg
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W1480020858