Contributions to Algebra and Geometry

38(1), 161 - 181 (1997)

Institut f"ur Mathematik, Universit"at Leipzig, Augustusplatz 10, D-04109 Leipzig, Germany. e-mail: kripfganz@mathematik.uni-leipzig.de

**Abstract:** In this paper pairs of plane figures inscriped into circular sectors and linked together by the fixed total perimeter length are considered. Among them, a pair of least total area and the corresponding perimeter partition are determined. The basis for this problem forms the generalized Favard problem discussed in the paper of A. Kripfganz published in Contributions to Algebra and Geometry 36(1995)2. The special case for figures of minimal area with fixed circumradius and perimeter length was studied by M. J. Favard in his paper 'ProblŠmes d'Extremums Relativs aux Courbes Convexes I', Annales Scientifiques de l'Ecole Normale Superieure 46(1929)3. The area of the optimal almost regular inpolyeder of the circumcircle depends on the perimeter length, parametrically. This optimal value function - the so-called 'fonction penetrante' - influences the solution structure of the perimeter partition problem which is discussed in dependence on the total perimeter length. For this, local and global methods of optimization theory are used. A certain convexity defect of the 'fonction penetrante' causes a solution branching between symmetric and nonsymmetric perimeter partitions. Corresponding branch points are determined numerically. Optimal solutions of the isoperimetric partition problem are given by pairs of almost regular inpolyeders of circular sectors. The asymptotic behavior of the branch points in dependence on the number of vertices is investigated.

**Keywords:** area-minimal inpolygon, isoperimetric problem, parametric optimization

**Classification (MSC91):** 52A40, 52A38, 90C90, 90C31

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