Projective metric geometry of tropical nuclei: gap matrices, event loci, and order chambers

TL;DR

This paper explores the projective metric geometry of tropical nuclei, revealing isometric properties, polyhedral structures, and algebraic-geometric relationships through a categorical framework involving the Isbell nucleus.

Contribution

It introduces a categorical construction of the Isbell nucleus to unify metric and polyhedral structures in tropical convexity, linking algebraic slack to geometric distances.

Findings

The row span and column span polytopes are isometric under a Hilbert projective metric.

The gap matrix entries correspond exactly to distances to inequality tight loci.

In the square case, the Chebyshev center's radius relates to the minimum cycle mean in a digraph.

Abstract

The tropical row span and column span of a real matrix are, from the polyhedral point of view, different objects living in different ambient spaces. These polytopes are known to be combinatorially isomorphic as polyhedral complexes; we prove that they are isometric under a Hilbert projective metric. We show that this isometry, along with a considerable amount of additional metric and polyhedral structure, is a direct consequence of a single categorical construction: the Isbell nucleus of the matrix, viewed as a profunctor enriched over the extended reals. The projective nucleus carries two canonical structures inherited from enrichment. The first is a Hilbert projective metric, with respect to which the Isbell conjugate maps are mutually inverse isometries -- this is the Isometry Theorem. The second is a polyhedral cell decomposition cut out by the Isbell inequalities, recovering the type decomposition of tropical convexity. These two structures are linked pointwise by the \emph{gap matrix}. The Events Theorem identifies each positive entry of the gap matrix with the exact projective distance to the locus where the corresponding inequality becomes tight: algebraic slack in the Isbell inequalities equals geometric distance to the cell walls. Thresholding the gap matrix at successive radii produces a constructible sheaf of formal concept lattice towers, extracting discrete algebraic structure from the continuous geometry at each point. In the square case there is generically a unique full-dimensional cell. The Centering Theorem identifies its Chebyshev center -- the point maximally insulated from all cell walls -- and shows that the optimal radius equals the minimum directed cycle mean of an associated digraph, connecting the projective geometry of the nucleus to the classical theory of optimal assignments.