The logarithmic capacity of a compact set in the complex plane is given by
(1)
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where
(2)
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and runs over each probability measure on . The quantity is called the Robin's constant of and the set is said to be polar if or equivalently, .
The logarithmic capacity coincides with the transfinite diameter of ,
(3)
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If is simply connected, the logarithmic capacity of is equal to the conformal radius of . Tables of logarithmic capacities have been calculated (e.g., Rumely 1989).