Operations on constructible functions and generalized valuations

TL;DR

This paper proves that Alesker's generalized valuation operations coincide with sheaf-theoretic ones on constructible functions under mild conditions, extending classical integral geometry formulas to broader geometric objects.

Contribution

It establishes the equivalence of two approaches to valuations on constructible functions, confirming conjectural aspects and extending integral geometry results.

Findings

Operations on constructible functions and generalized valuations coincide under mild transversality.

Extended additive kinematic formulas from convex bodies to subanalytic sets.

Derived new kinematic formulas on the 3-sphere.

Abstract

Alesker's theory of generalized valuations unifies smooth measures and constructible functions on real analytic manifolds, extending classical operations on functions and measures. Alesker showed that these operations agree with the sheaf-theoretic ones on constructible functions under restrictive assumptions, leaving key aspects conjectural. In this article, we close this gap by proving that the two approaches indeed coincide on constructible functions under mild transversality assumptions. Our proof is based on a comparison with the corresponding operations on characteristic cycles. As applications, we extend additive kinematic formulas from convex bodies to compact subanalytic sets in Euclidean spaces and derive new kinematic formulas on the 3-sphere.