The FermatTorricelli problem in the case of three-point sets in normed planes Daniil A. Ilyukhin

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The Fermat–Torricelli problem in the case of three-point

sets in normed planes

Daniil A. Ilyukhin

Abstract

In the paper the Fermat–Torricelli problem is considered. The problem asks a point min-

imizing the sum of distances to arbitrarily given points in d-dimensional real normed spaces.

Various generalizations of this problem are outlined, current methods of solving and some

recent results in this area are presented. The aim of the article is to ﬁnd an answer to the

following question: in what norms on the plane is the solution of the Fermat–Torricelli problem

unique for any three points. The uniqueness criterion is formulated and proved in the work, in

addition, the application of the criterion on the norms set by regular polygons, the so-called

lambda planes, is shown.

1 Introduction

The problem of ﬁnding a point that minimizes the sum of distances from it to a given set of

points in a metric space was ﬁrst mentioned in the 17th century. In 1643 Fermat posed a problem

for three points on the Euclidean plane, and in the same century Torricelli proposed a solution to

this problem ([4]).

Since then, various generalizations of this problem have been considered. The problem was

formulated for an arbitrary number of points, the dimension of the space, as well as the norm

given in this space. The simplicity of the formulation allows us to consider the problem even in

an arbitrary metric space. For example, the problem for four points on the Euclidean plane was

solved by D. Fagnano ([3], [16]). And for the case of ﬁve points, it was proved that the problem is

unsolvable in radicals, the proof is given in [1] and [6]. In addition, there is a generalized problem

in which the vertices are considered together with some positive values, called weights. You can

read about the development of the weighted problem in the works [21], [22], [19]. In particular, the

existence and uniqueness of the solution of such a problem for three points on the Euclidean plane

were proved ([16]).

In 1934 Jarnik and Kessler [15] posed the problem of ﬁnding a graph of minimum length con-

necting a ﬁnite number of points in the Euclidean plane, and in 1941 it was discussed in Courant

and Robbins’s book “What is Mathematics?“ ([7]), the authors of which named the problem under

consideration Steiner’s problem in honor of Jakob Steiner. The theory of the Steiner problem is

presented in the monographs [9] and [14], and in the book “Theory of Extreme Networks“ ([13])

Ivanov and Tuzhilin proposed various generalizations of the problem and new lines of research. You

can read about the study of extremal networks in normed spaces in [17], [2].

This article will consider the classic version of the problem: ﬁnding a point for which the

minimum sum of distances to elements of a subset of a metric space is reached. We will call such

arXiv:2210.04374v1 [math.MG] 9 Oct 2022

Figure 1: The design proposed by Torricelli. The point Tis the solution of the problem for the

given points A, B, C ([20])

a formulation generalized Fermat–Torricelli problem (or simply Fermat–Torricelli problem). The

work is based on the article [18], which describes the application of a geometric approach to the

problem and presents some new results that are obtained in the framework of real ﬁnite-dimensional

normed spaces, called Minkowski spaces. In particular, the authors give the following result:

Theorem 1.1 ([18]).In Minkowski space, the solution of the Fermat–Torricelli problem is a sin-

gleton for any ﬁnite set of non-collinear points if and only if the norm is strictly convex.

The purpose of this work is to solve a problem with a weaker condition: we will look for norms

on the plane in which the solution is unique for any three-point boundaries. In this case, new

examples of norms appear that have the property of uniqueness of the solution for any set, for

example, the Manhattan plane.

The second section presents the main deﬁnitions, and also describes the geometric method

for ﬁnding a solution to the Fermat–Torricelli problem. Section 3 gives an answer to the question

posed: a proof of the uniqueness criterion is given. The last part is devoted to applying the resulting

criterion to the norms given by regular polygons.

I would like to express my gratitude to my scientiﬁc adviser, Doctor of Physical and Mathemat-

ical Sciences Professor A.A.Tuzhilin and Doctor of Physical and Mathematical Sciences Professor

A.O.Ivanov for posing the problem and constant attention to the work.

2 Basic deﬁnitions and preliminary results

All statements in this section will be formulated for the Minkowski space, so this will not be

speciﬁed.

Deﬁnition 2.1. A point x0is called a Fermat–Torricelli point for points A={x1, . . . , xn}if x=x0

minimizes Pn

i=1 |xxi|. The set of all such points will be denoted by ft(A).

From the properties of the function Pn

i=1 |xxi|, one can obtain the following assertion about the

set of solutions of the Fermat–Torricelli problem, see for example [5].

Proposition 2.2. Let A={x1,. . . , xn}be points in space. Then ft(A)is a non-empty, compact

and convex set.

We give two examples of solving the Fermat–Torricelli problem on a plane. The ﬁgure 2 shows

the vertices of an equilateral triangle, ﬁrst in the Euclidean plane, and then in the norm given

by a regular hexagon. In the ﬁrst case, the set of solutions contains a single point constructed in

such a way that the angles between the rays coming out of it in the direction of the vertices of the

triangle are equal. In the second case, we specify the location of the points of the given set: let

one of them be at the origin, and the other two — at neighboring vertices of the unit circle. Under

such conditions, the set of solutions will include all points of the constructed triangle, including the

interior and boundaries.

Figure 2: Examples of solutions to the Fermat–Torricelli problem on the Euclidean plane and on

the λ-plane

Now the geometric method for constructing the solution of the Fermat–Torricelli problem will

be presented. To use it, along with the original space X, one must consider its dual space X∗,

which consists of linear functionals.

Deﬁnition 2.3. A functional ϕ∈X∗is called norming for a vector x∈Xif kϕk= 1 and

ϕ(x) = kxk.

It is easy to see that the elements of the space Xobtained by multiplying the vector xby a

positive number khave the same set of norming functionals. That is, the set of such functionals

can be described using points of the unit sphere. Consider norming functionals on some normed

plane.

The ﬁgure 3 shows a section of the unit circle Scontaining both a ﬂattened and a smooth section.

Let us construct norming functionals for vectors starting at zero and ending at a point lying on S.

At the point zthe unit circle has a single support line, then for the corresponding vector there is

a unique norming functional ϕ4, its level line coincides with this support line. Internal ﬂattening

points xy also correspond to vectors with a unique norming functional. At the point xthe unit

circle has more than one reference line. Each of them deﬁnes a norming functional, that is, for a

vector with an end in x, there are inﬁnitely many norming functionals. The functionals ϕ1and ϕ2

are given by the limit positions of the reference line, while ϕ3is an arbitrary one.

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TheFermatTorricelliprobleminthecaseofthree-pointsetsinnormedplanesDaniilA.IlyukhinAbstractInthepapertheFermatTorricelliproblemisconsidered.Theproblemasksapointmin-imizingthesumofdistancestoarbitrarilygivenpointsind-dimensionalrealnormedspaces.Variousgeneralizationsofthisproblemareoutlined,currentm...

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The FermatTorricelli problem in the case of three-point sets in normed planes Daniil A. Ilyukhin

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