The Cauchy-Crofton formula and the Whitney arc property for definable sets

TL;DR

This paper demonstrates that definable cells in an o-minimal structure satisfy the Whitney arc property by applying the Cauchy-Crofton formula, linking geometric measure theory with o-minimal geometry.

Contribution

It establishes a new connection between the Cauchy-Crofton formula and the Whitney arc property for definable sets in o-minimal structures.

Findings

Definable cells bounded by rational-radius balls satisfy the Whitney arc property.

The Cauchy-Crofton formula can be effectively used in o-minimal geometry.

The result applies to structures extending the real numbers with additional definable sets.

Abstract

We use the Cauchy-Crofton formula to show that every definable cell (bounded by a ball with rational radius) in an O-minimal expansion of a field extension of the real numbers satisfies the Whitney arc property.