Kinematic formulas for area measures

TL;DR

This paper develops new integral geometric formulas for surface area measures of convex bodies, involving spherical Fourier transforms, and extends orthogonality relations to centered functions.

Contribution

It introduces principal kinematic and Crofton formulas involving linear operators on measures, connecting to spherical Fourier transforms and extending existing orthogonality results.

Findings

Derived principal kinematic and Crofton formulas for surface area measures.

Connected formulas to spherical Fourier transforms from Koldobsky's work.

Extended orthogonality relations to centered functions.

Abstract

We obtain a Principal Kinematic Formula and a Crofton Formula for surface area measures of convex bodies, both involving linear operators on the vector space of signed measures on the unit sphere $S^{d-1}$. These formulas are related to a localization of Hadwiger's Integral Geometric Theorem. The operators, mentioned above, will be shown to be compositions of spherical Fourier transforms originating in the work of Koldobsky. As an application of our Crofton Formula, we will find an extension of Koldobsky's orthogonality relation for such transforms from the case of even spherical functions to centered functions.