Potential Theory of the Farthest-Point Distance Function
نویسندگان
چکیده
We study the farthest-point distance function, which measures the distance from z ∈ 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 σE has many interesting properties that reflect the topology and geometry of the compact set E. We prove σE(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 σE(E) = 1 2 for smooth convex sets of constant width. We conjecture σE(E) ≤ 1 2 for all E.
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