tailieunhanh - Đề tài " An isoperimetric inequality for logarithmic capacity of polygons "

We verify an old conjecture of G. P´lya and G. Szeg˝ saying that the o o regular n-gon minimizes the logarithmic capacity among all n-gons with a fixed area. 1. Introduction The logarithmic capacity cap E of a compact set E in R2 , which we identify with the complex plane C, is defined by () − log cap E = lim (g(z, ∞) − log |z|), z→∞ where g(z, ∞) denotes the Green function of a connected component Ω(E) ∞ of C \ E having singularity at z = ∞; see [4, Ch. 7], [7, §]. By an n-gon with. | Annals of Mathematics An isoperimetric inequality for logarithmic capacity of polygons By Alexander Yu. Solynin and Victor A. Zalgaller Annals of Mathematics 159 2004 277 303 An isoperimetric inequality for logarithmic capacity of polygons By Alexander Yu. SoLyNiN and Victor a. Zalgaller Abstract We verify an old conjecture of G. Polya and G. Szego saying that the regular n-gon minimizes the logarithmic capacity among all n-gons with a fixed area. 1. Introduction The logarithmic capacity cap E of a compact set E in R2 which we identify with the complex plane C is defined by log cap E lim g z to log z where g z to denotes the Green function of a connected component Q E 3 to of C E having singularity at z to see 4 Ch. 7 7 . By an n-gon with n 3 sides we mean a simply connected Jordan domain Dn c C whose boundary dDn consists of n rectilinear segments called sides of Dn. A closed n-gon will be denoted by Dn. Our principal result is Theorem 1. For any polygon Dn having a given number of sides n 3 12 cap2 Dn cap2 Df ntan n n r2 1 1 n . Area Dn Area Df n24 nr2 1 2 1 n with the sign of equality only for the regular n-gons. In Theorem 1 and below T - denotes the Euler gamma function and Df stands for the regular n-gon centered at z 0 with one vertex at z 1. This paper was finalized during the first author s visit at the Technion - Israel Institute of Technology Spring 2001 under the financial support of the Lady Devis Fellowship. This author thanks the Department of Mathematics of the Technion for wonderful atmosphere and working conditions during his stay in Haifa. The research of the first author was supported in part by the Russian Foundation for Basic Research grant no. 00-01-00118a. 278 ALEXANDER YU. SOLYNIN AND VICTOR A. ZALGALLER In other words Theorem 1 asserts that the regular closed polygon has the minimal logarithmic capacity among all closed polygons with a fixed number of sides and prescribed area. For n 5 this solves an old problem posed by G. Polya

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