Functions on Okounkov bodies coming from geometric valuations (with an appendix by S\'ebastien Boucksom)

TL;DR

This paper investigates the topological behavior of functions on Okounkov bodies, highlighting conditions for their continuity and providing counterexamples to boundary continuity in general.

Contribution

It characterizes when functions on Okounkov bodies are continuous and presents an example demonstrating the failure of boundary continuity.

Findings

Functions are continuous on polyhedral Okounkov bodies.

Continuity along the boundary does not always hold.

Counterexample shows boundary discontinuity in general.

Abstract

We study topological properties of functions on Okounkov bodies as introduced by Boucksom-Chen and Witt-Nystr\"om. We note that they are continuous over the whole Okounkov body whenever the body is polyhedral, on the other hand, we exhibit an example that shows that continuity along the boundary does not hold in general.