Bounds on the $f$-Vectors of Tight Spans

TL;DR

This paper establishes bounds on the face numbers of tight spans of metrics, revealing their combinatorial complexity and duality with regular triangulations of hypersimplices.

Contribution

It provides the first known tight upper bounds and partial lower bounds for the face numbers of tight spans of metrics, connecting them to hypersimplex triangulations.

Findings

Derived tight bounds for face numbers of tight spans

Linked tight spans to dual regular triangulations of hypersimplices

Enhanced understanding of the combinatorial structure of tight spans

Abstract

The tight span $T_d$ of a metric $d$ on a finite set is the subcomplex of bounded faces of an unbounded polyhedron defined by~$d$. If $d$ is generic then $T_d$ is known to be dual to a regular triangulation of a second hypersimplex. A tight upper and a partial lower bound for the face numbers of $T_d$ (or the dual regular triangulation) are presented.