Beiträge zur Algebra und Geometrie / Contributions to Algebra and GeometryVol. 40, No. 1, pp. 53-65 (1999)

About an Inequality of Kubota for Plane Convex Figures

Anita Kripfganz

Institut für Mathematik, Universität Leipzig, Augustusplatz 10, 04109 Leipzig, Germany

Abstract: This paper is concerned with area-minimal isoperimetric and isodiametric plane convex figures. The corresponding extremal problem is a concave minimization problem in a complete metric space. Among the subset of those figures having maximal circumradius the optimal one is determined as solution of a perimeter partition problem. A certain kind of nonsymmetric inpolyeders of the Reuleaux triangle solves the problem. The corresponding optimal value function gives lower bounds for the area in terms of the perimeter and the diameter. These results allow a sharpening of Kubota's isoperimetric inequality for the considered set of convex figures.

Keywords: geometric inequalities, plane convex figures, isoperimetric and isodiametric problem, global minimization

Classification (MSC91): 52A40; 52A38, 90C90

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