Mixed curvature measures of translative integral geometry

TL;DR

This paper develops a new representation for mixed curvature measures of sets with positive reach, extending previous work to multiple sets and providing detailed analysis for convex polyhedra.

Contribution

It introduces a representation of mixed curvature measures based on generalized curvatures, applicable to multiple sets with positive reach, including convex polyhedra.

Findings

Representation of mixed curvature measures using generalized curvatures

Extension of formulas to more than two sets

Detailed analysis for convex polyhedra

Abstract

The curvature measures of a set $X$ with singularities are measures concentrated on the normal bundle of $X$, which describe the local geometry of the set $X$. For given finitely many convex bodies or, more generally, sets with positive reach, the translative integral formula for curvature measures relates the integral mean of the curvature measures of the intersections of the given sets, one fixed and the others translated, to the mixed curvature measures of the given sets. In the case of two sets of positive reach, a representation of these mixed measures in terms of generalized curvatures, defined on the normal bundles of the sets, is known. For more than two sets, a description of mixed curvature measures in terms of rectifiable currents has been derived previously. Here we provide a representation of mixed curvatures measures of sets with positive reach based on generalized curvatures. The special case of convex polyhedra is treated in detail.