[For the entire collection see Zbl 0615.00005.]
C. H. FitzGerald, B. Rodin and St. E. Warschawski [Trans. Am. Math. Soc. 287, 681-685 (1985; Zbl 0533.30025)] have shown that if a continuum C in \(| z| \leq 1\) subtends an angle \(\leq \pi\) at the origin then the harmonic measure of C at the origin is at least as large as the harmonic measure at the origin of an arc of the same angle on \(| z| =1\). The author gives a very simply proof of this result by the method of the extremal metric. The result evidently fails for angles sufficiently close to \(2\pi\). W. H. J. Fuchs [Bull. Lond. Math. Soc. 16, 493 (1984)] raised the problem of finding the greatest lower bound at the origin of a set in \(| z| \leq 1\) which meets every radius. In the case where the set is restricted to be a continuum the configurations among which a minimum will occur are characterized. With an appropriate normalization these consist of arcs starting from the point 1 with an arc on \(| z| =1\) of angle at least \(\pi\) followed by an arc on the trajectory of a certain quadratic differential to a point on the segment (0,1). This uses the author’s basic result appearing in Math. Z. 135, 279-283 (1974; Zbl 0274.30019)]. Recently Marshall and Sundberg have been able by a rather different method to identify the actual extremal configuration. The author now can do so by a very simple argument on the earlier lines.