A plasticity principle of convex quadrilaterals on a convex surface of bounded specific curvature - Mathematics > Optimization and ControlReport as inadecuate




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Abstract: We derive the plasticity equations for convex quadrilaterals on a completeconvex surface with bounded specific curvature and prove a plasticity principlewhich states that: Given four shortest arcs which meet at the weightedFermat-Torricelli point their endpoints form a convex quadrilateral and theweighted Fermat-Torricelli point belongs to the interior of this convexquadrilateral, an increase of the weight corresponding to a shortest arc causesa decrease of the two weights that correspond to the two neighboring shortestarcs and an increase of the weight corresponding to the opposite shortest arcby solving the inverse weighted Fermat-Torricelli problem for quadrilaterals ona convex surface of bounded specific curvature. Furthermore, we show aconnection between the plasticity of convex quadrilaterals on a complete convexsurface with bounded specific curvature with the plasticity of some generalizedconvex quadrilaterals on a manifold which is certainly composed by triangles.We also study some cases of symmetrization of weighted convex quadrilaterals byintroducing a new symmetrization technique which transforms some classes ofweighted geodesic convex quadrilaterals on a convex surface to parallelogramsin the tangent plane at the weighted Fermat-Torricelli point of thecorresponding quadrilateral.



Author: Anastasios Zachos

Source: https://arxiv.org/







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