Strictly Monotone Numerosity on Tame Sets via the Steiner Polynomial
Joseph T. Previdi
TL;DR
The work introduces a renormalized intrinsic volume polynomial $Φ$ on bounded definable sets in an o-minimal framework, defined by $Φ(A)(t)=\sum_{k=0}^{\dim A} μ_k(A) t^k$, omitting binomial coefficients to preserve multiplicativity under products and to enable end-behavior ordering. It proves a Hadwiger-type uniqueness: $Φ$ is the unique (up to positive scaling of $t$) conormal continuous, similarity-equivariant homomorphism from constructible functions to $\mathbb{R}[t]$, thereby ensuring strict monotonicity with respect to subset inclusion. The paper then builds a bridge to nonstandard analysis by constructing a numerosity $\mathfrak{n}$ on $\mathcal{P}(\mathbb{R}^d)$ that approximates $Φ(A)(ω)$ for a hyperinteger $ω$, enabling asymptotic counting that recovers $Φ$ on tame sets and extends to all subsets. This synthesis connects o-minimal intrinsic geometry with numerosity theory, showing that intrinsic geometric information can be realized as a hyperfinite counting object and offering a unified perspective on dimension, measure, and monotonicity.
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} $Φ$, 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: $Φ$ 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.