Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Vol. 39, No. 2, pp. 447-459 (1998)

Solution Branching of a General Perimeter Partition Problem

Anita Kripfganz

Institut für Mathematik, Universität Leipzig, Augustusplatz 10, D-04109 Leipzig, Germany, e-mail:

Abstract: In this paper, a special kind of global minimization problems is solved and applied to a geometrical extremal problem. The geometrical problem deals with area-minimal m-tupel of plane figures. Each of the figures is inscriped into the same unit circular sector and called sector indomain. The total perimeter length of these m-tupel is parametrically fixed.

This isoperimetric problem shows a hierarchic structure. It is based on the generalized Favard problem (for one sector indomain) and the perimeter partition problem for pairs of sector indomains. These problems are investigated by the author in articles published in this journal, vols. 36 (1995) (2) and 38 (1997) (1). The corresponding results allow to formulate the geometrical problem as a nonlinear parametric optimization problem. Its solution structure is determined using the method of complete induction. In dependence on the given total perimeter length symmetric and non-symmetric perimeter partitions are optimal, resp. Corresponding branch points are numerically calculated for different sector angles and perimeters.

Keywords: isoperimetric problem, global optimization, parametric optimization, convexity defect

Classification (MSC91): 52A4052A38, 90C90, 90C31

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