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| Format: | Preprint |
| Published: |
2020
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2008.09969 |
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Table of Contents:
- 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.