Solving a Generalized Gauss Problem

Abstract

We solve a generalized Gauss problem in the Euclidean plane which states that: Given a convex quadrilateral, a positive number (weight) that corresponds to each of its vertices and a length of a linear segment which connects two mobile interior points of the quadrilateral find the minimum weighted network, which connects two of the vertices with one interior point and the other two with another interior point (Generalized Gauss tree). Furthermore, we introduce a generalized Gauss variable which corresponds to the unknown weight of the given distance which connects the two mobile interior points and obtain a degenerate generalized Gauss tree which corresponds to a specific value of the generalized Gauss variable that minimizes the length of the induced generalized Gauss trees and the weighted Fermat–Torricelli tree for a specific value of the generalized Gauss variable that maximizes the length of the induced generalized Gauss trees. Following this technique, we introduce a new class of generalized Gauss trees that we call absorbing generalized Gauss trees and a new class of Fermat–Torricelli trees that we call absorbing Fermat–Torricelli trees with respect to the sum of the four given weights of the convex quadrilateral.