Geometric Steiner Problems

Abstract

Historically, a geometric version of the Steiner problem in graphs was the first Steiner problem to be considered. It dates back to Fermat (1601–1665) who proposed the following problem: “Given three points in the plane, find a fourth point such that the sum of its distances to the three given points is minimum.” It is known that Torricelli found a geometric solution to Fermat’s problem before 1640. The generalization of Fermat’s problem to n instead of 3 points, that is finding a point minimizing the sum of the distances to n given points, was studied by many researchers. One of them was Jacob Steiner (1796–1863), a professor for geometry at the University of Berlin. It was him to whom Courant and Robbins in their book “What is Mathematics” attributed in 1941 a problem nowadays known as the Euclidean Steiner Problem: Given n points in the plane, find a shortest network which interconnects them. The two mathematicians Jarník and Kössler seem to have been the first who seriously considered this problem around 1930. Whether Jacob Steiner was even aware of it is unknown.