Higher rank inner products, Voronoi tilings and metric degenerations of tori

TL;DR

This paper introduces higher rank inner products and explores their Voronoi tilings, applying the framework to analyze metric degenerations of polarized tori and limits of Voronoi tilings.

Contribution

It develops a new framework of higher rank inner products and applies it to the study of metric degenerations and Voronoi tilings of tori.

Findings

Higher rank inner products generalize classical inner products.

Voronoi tilings are characterized for these new inner products.

Results describe limits of Voronoi tilings under metric degenerations.

Abstract

We introduce higher rank inner products on real and complex vector spaces and study their corresponding Voronoi tilings. We use the framework to describe metric degenerations of polarized tori and Hausdorff limits of Voronoi tilings of discrete sets.