Chebyshev centers and radius of the set of permutons

TL;DR

This paper determines the Chebyshev radius of the set of permutons under the rectangular distance $d_{ ectangle}$ and characterizes all Chebyshev centers as exactly the $1/2$-periodic permutons, yielding a precise geometric description of the space $ ext{Perm}$. The authors develop a toric reformulation and measure-theoretic toolkit, culminating in a key lemma linking $1/2$-periodicity to several rectangle-based conditions and proving that the diameter is $1/2$ while the radius is $1/4$. A stability theorem connects the enclosure of $ ext{Perm}$ within a slightly enlarged ball to proximity to a $1/2$-periodic center. They also classify and construct witnesses achieving the extremal distance, distinguishing universal (trivial) witnesses from nontrivial ones and highlighting a flat, robust geometry of permuton space.

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.

Figures (1)

• Figure 1: Rectangles in the Chebyshev-center argument.

Theorems & Definitions (17)

• Theorem 1.1
• Remark 1.2
• Lemma 2.1
• proof
• Proposition 2.2
• proof
• Theorem 2.3: Folland
• Lemma 3.1
• proof
• proof : Proof of Theorem \ref{['thm:main']}