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# Potential Theory of the Farthest-Point Distance Function

Published:2003-09-01
Printed: Sep 2003
• Richard S. Laugesen
• Igor E. Pritsker
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## Abstract

We study the farthest-point distance function, which measures the distance from $z \in \mathbb{C}$ 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 $\sigma_E$ has many interesting properties that reflect the topology and geometry of the compact set $E$. We prove $\sigma_E(E) \leq \frac12$ 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 $\sigma_E(E) = \frac12$ for smooth convex sets of constant width. We conjecture $\sigma_E(E) \leq \frac12$ for all~$E$.
 Keywords: distance function, farthest points, subharmonic function, representing measure, convex bodies of constant width
 MSC Classifications: 31A05 - Harmonic, subharmonic, superharmonic functions 52A10 - Convex sets in $2$ dimensions (including convex curves) [See also 53A04] 52A40 - Inequalities and extremum problems

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