Strictly Monotone Numerosity on Tame Sets via the Steiner Polynomial

TL;DR

This paper introduces a new valuation called the intrinsic volume polynomial, which is strictly monotone on tame sets and connects integral geometry with nonstandard analysis, leading to a unique algebraic characterization.

Contribution

It defines the intrinsic volume polynomial as a normalized Steiner polynomial, proving its strict monotonicity and uniqueness, and establishes a link between integral geometry and numerosity theory.

Findings

The intrinsic volume polynomial is strictly monotone on tame sets.

Uniqueness of the valuation as a conormal continuous, similarity-equivariant homomorphism.

Existence of a numerosity approximating the intrinsic volume polynomial.

Abstract

This paper uses inspiration from Integral Geometry to connect Tame Geometry with Nonstandard Analysis. We omit binomial coefficients from the Steiner polynomial to define the \textit{intrinsic volume polynomial} $\Phi$, a valuation defined on bounded definable sets in an o-minimal structure. We prove that using this normalization gives a strictly monotone valuation on point sets when the codomain $\mathbb{R}[t]$ is interpreted with ordering by end behavior. This leads to an algebraic version of Hadwiger's Theorem: $\Phi$ is the unique conormal continuous, similarity-equivariant homomorphism of ordered rings from $\mathcal{C}(\mathbb{R}^\infty) \to \mathbb{R}[t]$ (up to scaling). Noting that strict monotonicity is mirrored in numerosity theory (a branch of nonstandard analysis), we prove existence for a numerosity that exceptionally approximates the intrinsic volume polynomial. This suggests a connection between disparate fields, allowing each to complement the other.