Potential Theory of the Farthest-Point Distance Function

We study the farthest-point distance function, which measures the distance from z ε ℂ to the farthest point or points of a given compact set E in the plane. The logarithm of this distance is subharmonic as a function of z, and equals the logarithmic potential of a unique probability measure with unbounded support. This measure σ_E has many interesting properties that reflect the topology and geometry of the compact set E. We prove σ_E(E) ≤ 1/2 for polygons inscribed in a circle, with equality if and only if E is a regular n-gon for some odd n. Also we show σ_E(E) = 1/2 for smooth convex sets of constant width. We conjecture σ_E(E) ≤ 1/2 for all E.