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
azachos@gmail.com
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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