Hausdorff measure on o-minimal structures

TL;DR

This paper develops a Hausdorff measure theory within o-minimal structures, establishing key geometric formulas and properties for definable sets, including rectifiability and the Whitney arc property.

Contribution

It introduces the Hausdorff measure for definable sets in o-minimal structures and proves fundamental geometric formulas and rectifiability results.

Findings

Proved Cauchy-Crofton and co-area formulas for o-minimal Hausdorff measure

Showed every definable set can be partitioned into basic rectifiable sets

Established Whitney arc property for these sets

Abstract

We introduce the Hausdorff measure for definable sets in an o-minimal structure, and prove the Cauchy-Crofton and co-area formulae for the o-minimal Hausdorff measure. We also prove that every definable set can be partitioned into "basic rectifiable sets", and that the Whitney arc property holds for basic rectifiable sets.