On valuation deviations and their geometry

TL;DR

This paper introduces valuation-based deviations on convex bodies and explores their geometric properties through intrinsic pseudometrics, revealing insights into the underlying valuations.

Contribution

It presents a novel framework linking valuation deviations with geometric structures via intrinsic pseudometrics, advancing understanding of valuation geometry.

Findings

Established relationships between valuation deviations and convex body geometry.

Developed intrinsic pseudometrics to analyze valuation-based deviations.

Provided new geometric insights into valuation theory.

Abstract

We introduce two valuation-based deviations on convex bodies. Using a construction that allows us to associate to these deviations "intrinsic" pseudometrics, we establish various results which capture information about the underlying valuation in terms of the geometry of their induced deviations.