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Journal of Convex Analysis 21 (2014), No. 3, 619--650
Copyright Heldermann Verlag 2014

Minimal Systolic Circles

Anastasios N. Zachos
Dept. of Mathematics, University of Patras, 26500 Rion, Greece

We control the evolution of convex cyclic polygons by calculating the corresponding evolutionary circumradius (minimal systolic circle) each time a convex polygon is inscribed to a circle until it reaches the termination circle (minimum systolic circle) of the isoperimetric problem. We show that there exists a minimal circumradius for weighted convex quadrilaterals and pentagons such that their sides are given by the variable weights which satisfy the isoperimetric condition of the corresponding inverse weighted Fermat-Torricelli problem and the dynamic plasticity equations in the two dimensional Euclidean space. By splitting the weights along the prescribed rays which meet at the corresponding weighted Fermat-Torricelli point we deduce the generalized plasticity equations for convex polygons and we show that for a large number of variable weights the minimal circumradius approaches the minimum circumradius which corresponds to a regular polygon for equal weights. Furthermore, we obtain that the Gauss' minimal systolic circle of the generalized Gauss problem is smaller than the Fermat's minimal systolic circle of the Fermat-Torricelli problem for convex quadrilaterals.

Keywords: Isoperimetric inequality, polygons, dynamic plasticity, generalized plasticity, inverse Fermat-Torricelli problem, systolic circle, Fermat-Torricelli problem, Gauss problem.

MSC: 52A40, 51M16, 51N20

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