Valuations on manifolds and integral geometry

TL;DR

This paper develops new valuation operations on manifolds, introduces a generalized Radon transform encompassing classical cases, and provides an inversion formula in specific geometric settings, advancing integral geometry theory.

Contribution

It introduces new pull-back and push-forward operations on valuations, defines a general Radon transform, and connects these to classical integral geometry concepts.

Findings

Generalized Radon transform on valuations is introduced.

Inversion formula established for real projective spaces.

Connections made to Crofton and kinematic formulas.

Abstract

One constructs new operations of pull-back and push-forward on valuations on manifolds with respect to submersions and immersions. A general Radon type transform on valuations is introduced using these operations and the product on valuations. It is shown that the classical Radon transform on smooth functions, and the well known Radon transform on constructible functions with respect to the Euler characteristic are special cases of this new Radon transform. An inversion formula for the Radon transform on valuations has been proven in a specific case of real projective spaces. Relations of these operations to yet another classical type of integral geometry, Crofton and kinematic formulas, are indicated.