Geodesic
Potential Theory of the Farthest-Point Distance Function
Canadian mathematical bulletin, Tome 46 (2003) no. 3, pp. 373-387

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DOI

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 $E$ .
DOI : 10.4153/CMB-2003-039-0
Mots-clés : 31A05, 52A10, 52A40, distance function, farthest points, subharmonic function, representing measure, convex bodies of constant width 
Laugesen, Richard S.; Pritsker, Igor E. Potential Theory of the Farthest-Point Distance Function. Canadian mathematical bulletin, Tome 46 (2003) no. 3, pp. 373-387. doi: 10.4153/CMB-2003-039-0
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     title = {Potential {Theory} of the {Farthest-Point} {Distance} {Function}},
     journal = {Canadian mathematical bulletin},
     pages = {373--387},
     year = {2003},
     volume = {46},
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     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2003-039-0/}
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