Minimal Systolic Circles

A. N. Zachos

Journal of convex analysis, Tome 21 (2014) no. 3, pp. 619-65

Résumé

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.

Détail

Classification : 52A40, 51M16, 51N20
Mots-clés : Isoperimetric inequality, polygons, dynamic plasticity, generalized plasticity, inverse Fermat-Torricelli problem, systolic circle, Fermat-Torricelli problem, Gauss problem