The Hausdorff distance and metrics on toric singularity types

TL;DR

This paper compares the pseudo-metric on singularity types of Kähler manifolds with the Hausdorff metric, establishing H"older bounds and analyzing how geometric properties of convex bodies influence these bounds.

Contribution

It introduces a quasi-metric on convex sets, relates it to the existing pseudo-metric on singularity types, and provides optimal H"older bounds depending on the geometry of the convex body.

Findings

The topologies induced by the pseudo-metric and Hausdorff metric are equivalent.

H"older bounds are optimal and depend on the convex body's geometry.

Worst bounds occur for polytopes; best bounds for smooth convex bodies.

Abstract

Given a compact K\"ahler manifold $(X,\omega)$, due to the work of Darvas-Di Nezza-Lu, the space of singularity types of $\omega$-psh functions admits a natural pseudo-metric $d_\mathcal S$ that is complete in the presence of positive mass. When restricted to model singularity types, this pseudo-metric is a bona fide metric. In case of the projective space, there is a known one-to-one correspondence between toric model singularity types and convex bodies inside the unit simplex. Hence in this case it is natural to compare the $d_\mathcal S$ metric to the classical Hausdorff metric. We provide precise H\"older bounds, showing that their induced topologies are the same. More generally, we introduce a quasi-metric $d_G$ on the space of compact convex sets inside an arbitrary convex body $G$, with $d_\mathcal S = d_G$ in case $G$ is the unit simplex. We prove optimal H\"older bounds comparing $d_G$ with the Hausdorff metric. Our analysis shows that the H\"older exponents differ depending on the geometry of $G$, with the worst exponents in case $G$ is a polytope, and the best in case $G$ has $C^2$ boundary.