A new tropical volume, called the tropical barycentric volume, is defined from a tropical Ehrhart polynomial and is shown to satisfy the standard volume axioms and to be upper bounded by the dequantized tropical volume.
Tropical bisectors and Voronoi diagrams
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abstract
In this paper we initiate the study of tropical Voronoi diagrams. We start out with investigating bisectors of finitely many points with respect to arbitrary polyhedral norms. For this more general scenario we show that bisectors of three points are homeomorphic to a non-empty open subset of Euclidean space, provided that certain degenerate cases are excluded. Specializing our results to tropical bisectors then yields structural results and algorithms for tropical Voronoi diagrams.
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2019 1verdicts
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Tropical Ehrhart Theory and Tropical Volume
A new tropical volume, called the tropical barycentric volume, is defined from a tropical Ehrhart polynomial and is shown to satisfy the standard volume axioms and to be upper bounded by the dequantized tropical volume.