Maximum of the fourth diameter in the family of continua with prescribed capacity

G. V. Kuz'mina

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 9, Tome 59 (1976), pp. 60-80

Résumé

We obtain a complete solution of the problem of the maximum of the fourth diameter $$ d_4(E)=\biggl\{\max_{z_k,z_r\in E}\prod_{1\leqslant k\leqslant l\leqslant4}|z_k-z_l|\biggr\}^{1/6} $$ in the family of continua with capacity 1. Let $E(0,e^{i\alpha},e^{-i\alpha})$, $0\alpha\pi/2$, be a continuum of minimum capacity containing the points $0$, $e^{i\alpha}$, $e^{-i\alpha}$; $H(\alpha)=\operatorname{cap}E(0,e^{i\alpha},e^{-i\alpha})$. Let $c(\alpha)$ be the common point of three analytic arcs which form $E(0,e^{i\alpha},e^{-i\alpha})$. One shows that the indicated maximum is realized by the continuum $\mathscr E=\{z:H(\alpha_0)z^2\in E(0,e^{i\alpha},e^{-i\alpha})\}$ where $\alpha_0$, $0\alpha_0\pi/2$, is a solution of the equation $c(\alpha)=\frac13\cos\alpha$. Any other extremal continuum of the gives problem is an image of $\mathscr E$ under the mapping $z\to e^{i\gamma}z+C$ ($\gamma$ is a real and $C$ is a complex constant). One finds the value of the required maximum. The paper contains a brief exposition of the proof of this result.