Area, width, and logarithmic capacity of convex sets.
This paper continues the study of geometric properties of the logarithmic capacity cap\(E\) of a planar compact set \(E\); (see also the paper [Ann. Acad. Sci. Fenn. Math. 27, 419--436 (2002; Zbl 1017.30033)], by the same authors). The authors prove that if cap\,\(E=1\) and the projection of \(E\) on a straight line has length \(2h\) (\(0\leq h\leq 2\)), then the area of \(E\) has an upper bound in terms of \(h\). This bound involves elliptic integrals of the second kind. The authors give a complete description of the extremal configurations in terms of certain conformal maps. They apply this result to the case when \(E\) is convex and obtain an upper bound of the area of convex sets in terms of their logarithmic capacity and their width.
scientific article; zbMATH DE number 2101119