A motivic local Cauchy-Crofton formula

TL;DR

This paper develops a version of the local Cauchy-Crofton formula within the context of motivic integration for definable sets over Henselian discretely valued fields, enabling the computation of local densities through projections.

Contribution

It introduces a motivic local Cauchy-Crofton formula for definable sets in Henselian valued fields, linking local densities to projections over the Grassmannian.

Findings

Established a motivic local Cauchy-Crofton formula

Connected local density computation to projections over Grassmannian

Extended classical geometric measure theory to motivic setting

Abstract

In this note, we establish a version of the local Cauchy-Crofton formula for definable sets in Henselian discretely valued fields of characteristic zero. It allows to compute the motivic local density of a set from the densities of its projections integrated over the Grassmannian.