The plasticity of non-overlapping convex sets in R^{2}

TL;DR

This paper explores a generalized Fermat-Torricelli problem involving convex curves in the plane, introducing 'plasticity' solutions for non-overlapping circles with variable radii based on first variation formulas.

Contribution

It extends the classical problem to convex curves and introduces a novel 'plasticity' approach for circles with variable radii, providing new solutions and inverse problem analysis.

Findings

Derived first variation formula for line segment lengths.

Found 'plasticity' solutions for non-overlapping circles with variable radii.

Analyzed inverse problem related to the generalized Fermat-Torricelli problem.

Abstract

We study a generalization of the weighted Fermat-Torricelli problem in the plane, which is derived by replacing vertices of a convex polygon by 'small' closed convex curves with weights being positive real numbers on the curves, we also study its generalized inverse problem. Our solution of the problems is based on the first variation formula of the length of line segments that connect the weighted Fermat-Torricelli point with its projections onto given closed convex curves. We find the 'plasticity' solutions for non-overlapping circles with variable radius.