Chebyshev centers and radius of the set of permutons

TL;DR

This paper investigates the geometric structure of permutons under the rectangular distance, identifying the Chebyshev radius, characterizing centers, and describing extremal permutons.

Contribution

It determines the Chebyshev radius of permutons and characterizes all Chebyshev centers based on periodicity in each coordinate.

Findings

Chebyshev radius of permutons is 1/4

Chebyshev centers are permutons that are 1/2-periodic in each coordinate

Permutons at extremal distance 1/4 from a center are described

Abstract

We study the metric geometry of the set of permutons under the rectangular distance $d_{\square}$. We determine the Chebyshev radius to be 1/4 and characterize all Chebyshev centers: a permuton is a center if and only if it is 1/2- periodic in each coordinate. We also describe permutons that attain the extremal distance 1/4 from a given center.