Generalized Steiner’s Problem and its Solution with the Concepts in Field Thoery

We generalized the Steiner’s shortest line problem and found its connection with the concepts in classical field theory. We solved the generalized Steiner’s problem by introducing a conservative potential and a dissipative force in the field and gave a computing method by using a testing point and a corresponding iterative curve. 1 Steiner’s Problem In the R Euclidean space, there are n points αi(i = 1, ..., n.). Find a point αsteiner that the sum of its distances to the n points is a minimum: ∀α ∈ R, n i=1 |αsteiner − αi| ≤ n i=1 |α− αi|. (1) This is the original Steiner’s problem[1]. 2 Generalized Steiner’s Problem We would like to expand this problem to a more general situation: In the D-dimension differentiable linear space E, there are n points αi(i = 1, ..., n.). How to find a point αsteiner that the sum of its defined distances dis to the n points is a minimum: ∀α ∈ E, n i=1 dis(αsteiner − αi) ≤ n i=1 dis(α− αi). (2) We call the point αsteiner as the Steiner point. 3 Mapping Concepts onto Field Theory If we map the concept of ’distance’ in the generalized Steiner’s problem onto the concept of ’force potential’ in the Filed Theory, we will obtain another equivalent form of the generalized problem. Let’s consider that a particle at the testing point has conservative