Chebyshev centers and radius of the set of permutons

We study the metric geometry of the set of permutons under the rectangular distance $d_{\mathrm{\square }}$. 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.