The transfinite diameter of a nonempty compact set in the complex plane is , where the th diameter is the largest geometric mean of pairwise distances among points in . For ,

The points may repeat, so the formula also applies to finite . The numbers decrease nonstrictly to the logarithmic capacity of (Ransford 1995).

If the connected component of the complement of in the Riemann sphere containing the point at infinity is simply connected and , the unique normalized conformal mapping from the exterior of the unit disk onto this domain has the Laurent series

with and . The coefficient is the exterior conformal radius, and the corresponding Robin constant is (Ransford 1995).

The transfinite diameter also bounds the number of connected components of the open set when all the roots of a monic polynomial of polynomial degree lie in . Ghosh and Ramachandran (2024) proved that if , there is an depending on such that this number is at most for all sufficiently large . If is a connected set and , the open set has exactly one connected component for every .

At capacity 1, the line segment lies in no closed disk of radius 1. The monic polynomial has all its roots in , where is the Chebyshev polynomial of the first kind. The corresponding open set has connected components for every . This answers the existential reading of problem 1042(i) in the Erdős problems collection (Ghosh and Ramachandran 2024). It remains open whether every compact set of capacity 1 that lies in no closed disk of radius 1 permits connected components for infinitely many .

See also

Analytic Function, Logarithmic Capacity, Regular Function, Univalent Function

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References

Ghosh, S. and Ramachandran, K. "Number of Components of Polynomial Lemniscates: A Problem of Erdős, Herzog, and Piranian." J. Math. Anal. Appl. 540, 128571, 2024. https://doi.org/10.1016/j.jmaa.2024.128571.Ransford, T. Potential Theory in the Complex Plane. Cambridge, England: Cambridge University Press, 1995.

Referenced on Wolfram|Alpha

Transfinite Diameter

Cite this as:

Weisstein, Eric W. "Transfinite Diameter." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TransfiniteDiameter.html

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