Integer-valued valuations

TL;DR

This work addresses the classification of integer-valued monotone $\sigma$-continuous valuations on convex bodies without invariance assumptions, focusing on the planar and line cases. The authors prove that such valuations in dimensions $1$ and $2$ are representable as at most countable signed sums of Euler characteristics of intersections with locally finite convex sets, with a negative part admissible relative to the positive part; the planar case crucially relies on Eggleston's theorem to deduce polyconvexity of the valuation's support. A parallel 1D theory provides a clear, explicit structure, and the paper introduces countably generated valuations and defines a product operation that extends the Groemer/Alesker framework to non-smooth, non-translation-invariant settings. These results lay a rigorous foundation for discrete geometric valuations and open avenues for higher-dimensional generalizations and integral representations via locally finite measures.

Abstract

We obtain a complete characterization of planar monotone $σ$-continuous valuations taking integer values, without assuming invariance under any group of transformations. We further investigate the consequences of dropping monotonicity or $σ$-continuity and give a full classification of line valuations. We also introduce a construction of the product for valuations of this type.

Figures (3)

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Theorems & Definitions (26)

• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Definition 3.1
• Theorem 3.2
• Example 3.3
• Example 3.4