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On the subadditivity of the transfinite diameter

Published online by Cambridge University Press:  24 October 2008

Menahem Schiffer
Menahem Schiffer
Affiliation:
Einstein Institute of PhysicsThe Hebrew UniversityJerusalem
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Extract

Let M be a bounded and closed set of points in the complex z plane; d(M), a set-function which is of great importance in potential and function theory, may then be defined as follows. n points z1, z2, …, zn in M are so chosen that the product of the mutual distances

has the greatest possible value Then it can be proved that

exists. Thus the set-function d(M), named by Fekete the transfinite diameter of M, is defined.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 37 , Issue 4 , October 1941 , pp. 373 - 383
Copyright
Copyright © Cambridge Philosophical Society 1941

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References

† Fekete, M., ‘Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten’, Math. Z. 17 (1923), 228–49.CrossRefGoogle Scholar

‡ Loc. cit. p. 239; for a certain generalization of this relation see Szegö, G., ‘Bemerkungen zu einer Arbeit von Herrn Fekete…’, Math. Z. 21 (1924), 203–8.CrossRefGoogle Scholar

† For an elementary proof see Fekete, M., ‘Über die Verallgemeinerung der Picard-Landauschen und Picard-Schottkyschen Sätze…’, Math. Ann. 106 (1932), 598Google Scholar, note 12.

‡ Fekete, M., Relations between the transfinite diameter of an arc and its length, Sefer Magnes, Jerusalem (1938), pp. 401–13Google Scholar (Hebrew with an abstract in English).

§ Pólya, and Szegö, , ‘Über den transfiniten Durchmesser (Kapazitätskonstante) von ebenen und räumlichen Punktmengen’, J. reine angew. Math. 165 (1931), 4–49.Google Scholar

∥ See for example Kellog, , Foundations of potential theory (Berlin, 1929), p. 331.CrossRefGoogle Scholar

Pólya and Szegö, loc. cit. This property of c(M) follows elementarily from definition (2) of d(M) and the identity (3). This was pointed out by Prof. Fekete in a seminar on the transfinite diameter held at the Hebrew University in 1934. Compare also Nevanlinna, R., Eindeutige analytische Funktionen (Berlin, 1936), p. 120CrossRefGoogle Scholar, Hilfssatz 1.

† Löwner, K., ‘Über Extremumssätze bei der konformen Abbildung des Äusseren des Einheitskreises’, Math. Z. 3 (1919), 65–77.CrossRefGoogle Scholar

† Carathéodory, C., ‘Untersuchungen über die konformen Abbildungen von festen und veränderlichen Gebieten’, Math. Ann. 72 (1912), 107–14.CrossRefGoogle Scholar

† Carathéodory, C., ‘Über die Begrenzung einfach-zusammenhängender Gebiete’, Math. Ann. 73 (1913), 323–70.CrossRefGoogle Scholar

‡ Bieberbach, L., ‘Über die Koeffizienten derjenigen Potenzreihen, …’, Berl. Sitzungsber. (1916), pp. 940–55.Google Scholar