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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 
${{\sigma }_{E}}(E)\,\le \,\frac{1}{2}$ 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 
${{\sigma }_{E}}(E)\,=\,\frac{1}{2}$ for smooth convex sets of constant width. We conjecture ${{\sigma }_{E}}(E)\,\le \,\frac{1}{2}$ for all $E$.
