- Email: [email protected]
The Steiner ratio of high-dimensional Banach–Minkowski spaces
Discrete Applied Mathematics 138 (2004) 29 – 34
www.elsevier.com/locate/dam
The Steiner ratio of high-dimensional Banach–Minkowski spaces Dietmar Cieslik∗ Inst. fur Mathematik und Informatik, University of Greifswald, Jahnstr. 15a, Greifswald 17487, Germany Received 13 September 2000; received in revised form 4 April 2002; accepted 28 February 2003
Abstract The Steiner ratio is the greatest lower bound of the ratios of the Steiner Minimal Tree- by the Minimum Spanning Tree-lengths running over all .nite subsets of a metric space. We will discuss this quantity for .nite-dimensional Banach spaces of high dimension. Particularly, let the quantity Cd de.ned as the upper bound of the Steiner ratio of all d-dimensional Banach spaces, then lim Cd = lim md (B(2));
d→∞
d→∞
where md (B(2)) denotes the Steiner ratio of the d-dimensional Euclidean space. ? 2003 Elsevier B.V. All rights reserved. Keywords: Steiner minimal trees; Steiner ratio; Geometry of Banach spaces
1. Introduction The history of Steiner’s Problem started with Fermat [11] early in the 17th century and Gau; [15] in 1836. At .rst perhaps with the famous book What is Mathematics [6] by Courant and Robbins in 1941, this problem became popularized under the name of Steiner: For a given .nite set of points in a metric space .nd a network which connects all points of the set with minimal length. ∗
Tel.: +49-3834-864614; fax: +49-3834-864615. E-mail address: [email protected] (D. Cieslik).
0166-218X/$ - see front matter ? 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0166-218X(03)00267-1
30
D. Cieslik / Discrete Applied Mathematics 138 (2004) 29 – 34
Such a network must be a tree, which is called a Steiner Minimal Tree (SMT). It may contain vertices, called Steiner points, other than the points which are to be connected. Given a set of points, it is a priori unclear how many Steiner points one has to add, and where these points are located, in order to construct an SMT. A classical survey of Steiner’s Problem in the Euclidean plane was presented by Gilbert and Pollak in 1968 [16] and christened “Steiner Minimal Tree” for the shortest interconnecting network and “Steiner points” for the additional vertices. Over the years Steiner’s Problem has taken on an increasingly important role. Consequently, in the last three decades the investigations and, naturally, the publications about Steiner’s Problem have increased rapidly. Surveys are given by Cieslik [4], Hwang et al. [17] and Ivanov and Tuzhilin [18]. However, all investigations showed the great complexity of the problem, as well in the sense of structural as in the sense of computational complexity.1 In other terms: Observation I. In general, methods to .nd an SMT are hard in the sense of computational complexity or still unknown. On the other hand, a Minimum Spanning Tree (MST), this is a shortest tree interconnecting a .nite set of points without using Steiner points, can be found easily by simple and general applicable methods, cf. [2]. Observation II. It is easy to .nd an MST by an algorithm which is simple to realize and runs fast in all metric space. Solutions of Steiner’s problem depend essentially on the way in which the distances in space are determined. We consider .nite-dimensional Banach spaces de.ned in the following way: Ad denotes the d-dimensional aJne space with origin o. A convex and compact body B of the Ad centered in the origin o is called a unit ball, and induces a norm :=:B in the corresponding linear space by the so-called Minkowski functional: vB = inf {t ¿ 0 : v ∈ tB} for any v in Ad \ {o}, and oB = 0. On the other hand, let : be a norm in Ad , then B = {v ∈ Ad : v 6 1} is a unit ball in the above sense. A norm is completely determined by its unit ball and vice versa. Consequently, a Banach–Minkowski space is uniquely de.ned by an aJne space Ad and a unit ball B, abbreviated as Md (B).2 Md (B) is a complete metric linear space if we de.ne the metric by
(v; v ) = v − v B :
(1)
1 For instance Steiner’s Problem in the Euclidean plane and in the plane with rectilinear norm is NP-hard, [13,14]. 2 Essentially, every Banach–Minkowski space is a .nite-dimensional Banach space and vice versa. For the geometry of Banach–Minkowski spaces cf. [19].
D. Cieslik / Discrete Applied Mathematics 138 (2004) 29 – 34
31
Particularly, we consider spaces with p-norm, de.ned in the following way: For the points v = (x1 ; : : : ; xd ) and v = (y1 ; : : : ; yd ) of the Ad we de.ne the distance by 1=p d p |xi − yi | ; v − v B(p) = i=1
where p is a real not less than 1. The quantity LB (SMT for N ) : N ⊆ Md (B) is a .nite set ; (2) md (B) := inf LB (MST for N ) is called the Steiner ratio of Md (B). Given by Moore in [16] it holds that 1 1 ¿ md (B) ¿ : (3) 2 The Steiner ratio describes the performance ratio of the approximation for Steiner’s Problem by an MST. Moreover, there is a polynomial-time approximation scheme for Steiner’s Problem in any Banach–Minkowski space, Arora [1]. Considering Euclidean spaces, the Steiner ratio of the planar case is known: √ 3 m2 (B(2)) = = 0:86602 : : : ; (4) 2 given by Du and Hwang [9]. For each dimension d ¿ 2, at present, the exact value of the Steiner ratio is not yet known. Chung and Gilbert [3] gave an upper and Du [7] found a lower bound for Euclidean spaces of high dimension: 0:61582 : : : 6 lim md (B(2)) 6 0:66984 : : : : d→∞
(5)
For the Steiner ratio of several other spaces cf. [5]. 2. The Steiner ratio for Banach–Minkowski spaces of high dimensions There holds the following counterintuitive geometric assertion: Each unit ball in a suJciently large dimensional Banach space has a large almost ellipsoidal section. More exactly, we use the Banach–Mazur distance, which is a natural similarity measure for two Banach spaces of the same dimension, in the following way: Let Bd denote the class of all unit balls in Ad , and let [Bd ] be aJne equivalence classes for Bd . Then the Banach–Mazur distance L is a metric on [Bd ] de.ned as L([B]; [B ]) = ln inf {h ¿ 1: there is a bijective linear mapping A such that B ⊆ AB ⊆ hB}
(6)
for [B]; [B ] in [Bd ]. Remark 2.1 (Dvoretzky [10]). For each positive real number and each positive integer d there is a number D(; d ) such that every Banach–Minkowski space Md (B) of dimension d at least D(; d ) contains a d -dimensional subspace Md (B ) such that L([B ]; [B(2)]) 6 ln(1 + ):
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D. Cieslik / Discrete Applied Mathematics 138 (2004) 29 – 34
In terms of norms this fact means: For every positive integer d and every positive real there exists a number D(; d ) such that for every norm : in Ad , where d ¿ D(; d ), there exists a constant c ¿ 0 and a subspace Ad such that c · vB˜ 6 v 6 (1 + ) · c · vB˜
(7)
˜ is isometric to the d -dimensional Euclidean space. for all v ∈ Ad , where Md (B) Suppose that the assumption of Remark 2.1 is satis.ed and Md (B ) is the subspace of Md (B). Then we have,
md (B) 6 md (B ):
(8)
Moreover, the inequality L([B ]; [B(2)]) 6 ln(1 + )
(9)
implies (7). Then it is not hard to see that md (B ) 6 (1 + ) · md (B(2)):
(10)
Both, (8) and (10) give the following Lemma 2.2. For the positive integer d and the positive real number let D(; d ) be the Dvoretzky number, as de5ned in 2.1. Then for each Banach–Minkowski space Md (B) of dimension d at least D(; d ) the inequality md (B) 6 (1 + ) · md (B(2)) holds. 3. When the dimension runs to in$nity An interesting problem, but which seems very diJcult, is to determine the range of the Steiner ratio for d-dimensional Banach spaces, depending from the value d. More exactly, determine the best possible reals cd and Cd such that cd 6 md (B) 6 Cd
(11)
for all unit balls B of the d-dimensional aJne space. Both, the numbers Cd and cd , are attained by certain Banach spaces. This follows from the continuity of the Steiner ratio as a function of the space and the Blaschke selection theorem. The quantity Cd is de.ned as the upper bound of all numbers md (B) ranging over all unit balls B of the d-dimensional aJne space Ad : Cd = sup{md (B): B a unit ball in Ad }:
(12)
Consider the sequence {Cd }d=1; 2; ::: . Let Md (B ) be a (Banach–Minkowski) subspace of Md (B), d 6 d. Then it is not hard to see that md (B ) ¿ md (B): Hence, starting with C1 = 1 the sequence {Cd }d=1; 2; ::: is decreasing and bounded, consequently convergent.
D. Cieslik / Discrete Applied Mathematics 138 (2004) 29 – 34
33
Conjecture 3.1. For any number d = 1; 2; : : : the equality Cd = md (B(2)) holds. The conjecture is open, even for d = 2, we only know √ √ 3 13 − 1 6 C2 6 ; 2 3 in view of (4) and using a result by Du et al. [8]. Section 2.2 implies
(13)
md (B(2)) 6 Cd 6 (1 + ) · md (B(2)) 6 (1 + ) · Cd ; if d ¿ D(; d ). Suppose that d runs to in.nity, then d does as well. Hence, Theorem 3.2. Let the quantity Cd de5ned as the upper bound of all numbers md (B) ranging over all unit balls B of the d-dimensional a:ne space. Then {Cd }d=1; 2; ::: is a decreasing and convergent sequence with lim Cd = lim md (B(2)):
d→∞
d→∞
On the other hand, we are interested in cd = inf {md (B): B a unit ball in Ad }:
(14)
Here we give the Conjecture 3.3. For any number d = 1; 2; : : : the inequality Cd ¿ 0:5 holds. The conjecture is open, except for d = 2, where we know c2 = 23 ;
(15)
showed by Gao et al. [12]. In the d-dimensional aJne space, the unit ball B(1) is the convex hull of N = {±(0; : : : ; 0; 1; 0; : : : ; 0) : the ith component is equal to 1; i = 1; : : : ; d}: The set N contains 2d points. The distance of any two diOerent points in N equals 2. Hence, an MST for N has the length 2(2d − 1). An SMT for N with the Steiner point o has the length 2d. Together with (3) these facts imply d 1 (16) 2 6 md (B(1)) 6 2d − 1 : Consequently, Theorem 3.4. Let the quantity cd de5ned as the lower bound of all numbers md (B) ranging over all unit balls B of the d-dimensional a:ne space. Then {cd }d=1; 2; ::: is a convergent sequence with lim cd = 12 :
d→∞
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D. Cieslik / Discrete Applied Mathematics 138 (2004) 29 – 34
References [1] S. Arora, Polynomial time approximation schemes for Euclidean traveling salesman and other geometrical problems, J. Assoc. Comput. Mach. 45 (1998) 753–782. [2] D. Cheriton, R.E. Tarjan, Finding minimum spanning trees, SIAM J. Comput. 5 (1976) 724–742. [3] F.R.K. Chung, E.N. Gilbert, Steiner trees for the regular simplex, Bull. Inst. Math. Acad. Sinica 4 (1976) 313–325. [4] D. Cieslik, Steiner Minimal Trees, Kluwer Academic Publishers, Dordrecht, 1998. [5] D. Cieslik, The Steiner Ratio, Kluwer Academic Publishers, Dordrecht, 2001. [6] R. Courant, H. Robbins, What is Mathematics, Oxford University Press, New York, 1941. [7] D.Z. Du, On Steiner ratio conjectures, Ann. Oper. Res. 33 (1991) 437–449. [8] D.Z. Du, B. Gao, R.L. Graham, Z.C. Liu, P.J. Wan, Minimum Steiner trees in normed planes, Discrete Comput. Geom. 9 (1993) 351–370. [9] D.Z. Du, F.K. Hwang, A proof of the Gilbert–Pollak conjecture on the Steiner ratio, Algorithmica 7 (1992) 121–136. [10] A. Dvoretzky, Some results on convex bodies and Banach spaces, in: Proceedings of Symposium on Linear Spaces, Jerusalem, 1961, pp. 123–160. [11] P. Fermat, Abhandlungen uV ber Maxima und Minima, Oswalds Klassiker Exakt. Wiss. 238 (1934). [12] B. Gao, D.Z. Du, R.L. Graham, A tight lower bound for the Steiner ratio in Minkowski planes, Discrete Math. 142 (1995) 49–63. [13] M.R. Garey, R.L. Graham, D.S. Johnson, The complexity of computing Steiner minimal trees, SIAM J. Appl. Math. 32 (197) 835 –859. [14] M.R. Garey, D.S. Johnson, The rectilinear Steiner tree problem is NP-complete, SIAM J. Appl. Math. 32 (1977) 826–834. [15] C.F. Gau;, Briefwechsel Gau;-Schuhmacher, in: Werke Bd. X,1, GVottingen, 1917, pp. 459 – 468. [16] E.N. Gilbert, H.O. Pollak, Steiner minimal trees, SIAM J. Appl. Math. 16 (1968) 1–29. [17] F.K. Hwang, D.S. Richards, P. Winter, The Steiner tree problem, Ann. Discrete Math. 53 (1992). [18] A.O. Ivanov, A.A. Tuzhilin, Minimal Networks – The Steiner Problem and its Generalizations, CRC Press, Boca Raton, FL, 1994. [19] A.C. Thompson, Minkowski Geometry, Cambridge University Press, Cambridge, 1996.
www.elsevier.com/locate/dam
The Steiner ratio of high-dimensional Banach–Minkowski spaces Dietmar Cieslik∗ Inst. fur Mathematik und Informatik, University of Greifswald, Jahnstr. 15a, Greifswald 17487, Germany Received 13 September 2000; received in revised form 4 April 2002; accepted 28 February 2003
Abstract The Steiner ratio is the greatest lower bound of the ratios of the Steiner Minimal Tree- by the Minimum Spanning Tree-lengths running over all .nite subsets of a metric space. We will discuss this quantity for .nite-dimensional Banach spaces of high dimension. Particularly, let the quantity Cd de.ned as the upper bound of the Steiner ratio of all d-dimensional Banach spaces, then lim Cd = lim md (B(2));
d→∞
d→∞
where md (B(2)) denotes the Steiner ratio of the d-dimensional Euclidean space. ? 2003 Elsevier B.V. All rights reserved. Keywords: Steiner minimal trees; Steiner ratio; Geometry of Banach spaces
1. Introduction The history of Steiner’s Problem started with Fermat [11] early in the 17th century and Gau; [15] in 1836. At .rst perhaps with the famous book What is Mathematics [6] by Courant and Robbins in 1941, this problem became popularized under the name of Steiner: For a given .nite set of points in a metric space .nd a network which connects all points of the set with minimal length. ∗
Tel.: +49-3834-864614; fax: +49-3834-864615. E-mail address: [email protected] (D. Cieslik).
0166-218X/$ - see front matter ? 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0166-218X(03)00267-1
30
D. Cieslik / Discrete Applied Mathematics 138 (2004) 29 – 34
Such a network must be a tree, which is called a Steiner Minimal Tree (SMT). It may contain vertices, called Steiner points, other than the points which are to be connected. Given a set of points, it is a priori unclear how many Steiner points one has to add, and where these points are located, in order to construct an SMT. A classical survey of Steiner’s Problem in the Euclidean plane was presented by Gilbert and Pollak in 1968 [16] and christened “Steiner Minimal Tree” for the shortest interconnecting network and “Steiner points” for the additional vertices. Over the years Steiner’s Problem has taken on an increasingly important role. Consequently, in the last three decades the investigations and, naturally, the publications about Steiner’s Problem have increased rapidly. Surveys are given by Cieslik [4], Hwang et al. [17] and Ivanov and Tuzhilin [18]. However, all investigations showed the great complexity of the problem, as well in the sense of structural as in the sense of computational complexity.1 In other terms: Observation I. In general, methods to .nd an SMT are hard in the sense of computational complexity or still unknown. On the other hand, a Minimum Spanning Tree (MST), this is a shortest tree interconnecting a .nite set of points without using Steiner points, can be found easily by simple and general applicable methods, cf. [2]. Observation II. It is easy to .nd an MST by an algorithm which is simple to realize and runs fast in all metric space. Solutions of Steiner’s problem depend essentially on the way in which the distances in space are determined. We consider .nite-dimensional Banach spaces de.ned in the following way: Ad denotes the d-dimensional aJne space with origin o. A convex and compact body B of the Ad centered in the origin o is called a unit ball, and induces a norm :=:B in the corresponding linear space by the so-called Minkowski functional: vB = inf {t ¿ 0 : v ∈ tB} for any v in Ad \ {o}, and oB = 0. On the other hand, let : be a norm in Ad , then B = {v ∈ Ad : v 6 1} is a unit ball in the above sense. A norm is completely determined by its unit ball and vice versa. Consequently, a Banach–Minkowski space is uniquely de.ned by an aJne space Ad and a unit ball B, abbreviated as Md (B).2 Md (B) is a complete metric linear space if we de.ne the metric by
(v; v ) = v − v B :
(1)
1 For instance Steiner’s Problem in the Euclidean plane and in the plane with rectilinear norm is NP-hard, [13,14]. 2 Essentially, every Banach–Minkowski space is a .nite-dimensional Banach space and vice versa. For the geometry of Banach–Minkowski spaces cf. [19].
D. Cieslik / Discrete Applied Mathematics 138 (2004) 29 – 34
31
Particularly, we consider spaces with p-norm, de.ned in the following way: For the points v = (x1 ; : : : ; xd ) and v = (y1 ; : : : ; yd ) of the Ad we de.ne the distance by 1=p d p |xi − yi | ; v − v B(p) = i=1
where p is a real not less than 1. The quantity LB (SMT for N ) : N ⊆ Md (B) is a .nite set ; (2) md (B) := inf LB (MST for N ) is called the Steiner ratio of Md (B). Given by Moore in [16] it holds that 1 1 ¿ md (B) ¿ : (3) 2 The Steiner ratio describes the performance ratio of the approximation for Steiner’s Problem by an MST. Moreover, there is a polynomial-time approximation scheme for Steiner’s Problem in any Banach–Minkowski space, Arora [1]. Considering Euclidean spaces, the Steiner ratio of the planar case is known: √ 3 m2 (B(2)) = = 0:86602 : : : ; (4) 2 given by Du and Hwang [9]. For each dimension d ¿ 2, at present, the exact value of the Steiner ratio is not yet known. Chung and Gilbert [3] gave an upper and Du [7] found a lower bound for Euclidean spaces of high dimension: 0:61582 : : : 6 lim md (B(2)) 6 0:66984 : : : : d→∞
(5)
For the Steiner ratio of several other spaces cf. [5]. 2. The Steiner ratio for Banach–Minkowski spaces of high dimensions There holds the following counterintuitive geometric assertion: Each unit ball in a suJciently large dimensional Banach space has a large almost ellipsoidal section. More exactly, we use the Banach–Mazur distance, which is a natural similarity measure for two Banach spaces of the same dimension, in the following way: Let Bd denote the class of all unit balls in Ad , and let [Bd ] be aJne equivalence classes for Bd . Then the Banach–Mazur distance L is a metric on [Bd ] de.ned as L([B]; [B ]) = ln inf {h ¿ 1: there is a bijective linear mapping A such that B ⊆ AB ⊆ hB}
(6)
for [B]; [B ] in [Bd ]. Remark 2.1 (Dvoretzky [10]). For each positive real number and each positive integer d there is a number D(; d ) such that every Banach–Minkowski space Md (B) of dimension d at least D(; d ) contains a d -dimensional subspace Md (B ) such that L([B ]; [B(2)]) 6 ln(1 + ):
32
D. Cieslik / Discrete Applied Mathematics 138 (2004) 29 – 34
In terms of norms this fact means: For every positive integer d and every positive real there exists a number D(; d ) such that for every norm : in Ad , where d ¿ D(; d ), there exists a constant c ¿ 0 and a subspace Ad such that c · vB˜ 6 v 6 (1 + ) · c · vB˜
(7)
˜ is isometric to the d -dimensional Euclidean space. for all v ∈ Ad , where Md (B) Suppose that the assumption of Remark 2.1 is satis.ed and Md (B ) is the subspace of Md (B). Then we have,
md (B) 6 md (B ):
(8)
Moreover, the inequality L([B ]; [B(2)]) 6 ln(1 + )
(9)
implies (7). Then it is not hard to see that md (B ) 6 (1 + ) · md (B(2)):
(10)
Both, (8) and (10) give the following Lemma 2.2. For the positive integer d and the positive real number let D(; d ) be the Dvoretzky number, as de5ned in 2.1. Then for each Banach–Minkowski space Md (B) of dimension d at least D(; d ) the inequality md (B) 6 (1 + ) · md (B(2)) holds. 3. When the dimension runs to in$nity An interesting problem, but which seems very diJcult, is to determine the range of the Steiner ratio for d-dimensional Banach spaces, depending from the value d. More exactly, determine the best possible reals cd and Cd such that cd 6 md (B) 6 Cd
(11)
for all unit balls B of the d-dimensional aJne space. Both, the numbers Cd and cd , are attained by certain Banach spaces. This follows from the continuity of the Steiner ratio as a function of the space and the Blaschke selection theorem. The quantity Cd is de.ned as the upper bound of all numbers md (B) ranging over all unit balls B of the d-dimensional aJne space Ad : Cd = sup{md (B): B a unit ball in Ad }:
(12)
Consider the sequence {Cd }d=1; 2; ::: . Let Md (B ) be a (Banach–Minkowski) subspace of Md (B), d 6 d. Then it is not hard to see that md (B ) ¿ md (B): Hence, starting with C1 = 1 the sequence {Cd }d=1; 2; ::: is decreasing and bounded, consequently convergent.
D. Cieslik / Discrete Applied Mathematics 138 (2004) 29 – 34
33
Conjecture 3.1. For any number d = 1; 2; : : : the equality Cd = md (B(2)) holds. The conjecture is open, even for d = 2, we only know √ √ 3 13 − 1 6 C2 6 ; 2 3 in view of (4) and using a result by Du et al. [8]. Section 2.2 implies
(13)
md (B(2)) 6 Cd 6 (1 + ) · md (B(2)) 6 (1 + ) · Cd ; if d ¿ D(; d ). Suppose that d runs to in.nity, then d does as well. Hence, Theorem 3.2. Let the quantity Cd de5ned as the upper bound of all numbers md (B) ranging over all unit balls B of the d-dimensional a:ne space. Then {Cd }d=1; 2; ::: is a decreasing and convergent sequence with lim Cd = lim md (B(2)):
d→∞
d→∞
On the other hand, we are interested in cd = inf {md (B): B a unit ball in Ad }:
(14)
Here we give the Conjecture 3.3. For any number d = 1; 2; : : : the inequality Cd ¿ 0:5 holds. The conjecture is open, except for d = 2, where we know c2 = 23 ;
(15)
showed by Gao et al. [12]. In the d-dimensional aJne space, the unit ball B(1) is the convex hull of N = {±(0; : : : ; 0; 1; 0; : : : ; 0) : the ith component is equal to 1; i = 1; : : : ; d}: The set N contains 2d points. The distance of any two diOerent points in N equals 2. Hence, an MST for N has the length 2(2d − 1). An SMT for N with the Steiner point o has the length 2d. Together with (3) these facts imply d 1 (16) 2 6 md (B(1)) 6 2d − 1 : Consequently, Theorem 3.4. Let the quantity cd de5ned as the lower bound of all numbers md (B) ranging over all unit balls B of the d-dimensional a:ne space. Then {cd }d=1; 2; ::: is a convergent sequence with lim cd = 12 :
d→∞
34
D. Cieslik / Discrete Applied Mathematics 138 (2004) 29 – 34
References [1] S. Arora, Polynomial time approximation schemes for Euclidean traveling salesman and other geometrical problems, J. Assoc. Comput. Mach. 45 (1998) 753–782. [2] D. Cheriton, R.E. Tarjan, Finding minimum spanning trees, SIAM J. Comput. 5 (1976) 724–742. [3] F.R.K. Chung, E.N. Gilbert, Steiner trees for the regular simplex, Bull. Inst. Math. Acad. Sinica 4 (1976) 313–325. [4] D. Cieslik, Steiner Minimal Trees, Kluwer Academic Publishers, Dordrecht, 1998. [5] D. Cieslik, The Steiner Ratio, Kluwer Academic Publishers, Dordrecht, 2001. [6] R. Courant, H. Robbins, What is Mathematics, Oxford University Press, New York, 1941. [7] D.Z. Du, On Steiner ratio conjectures, Ann. Oper. Res. 33 (1991) 437–449. [8] D.Z. Du, B. Gao, R.L. Graham, Z.C. Liu, P.J. Wan, Minimum Steiner trees in normed planes, Discrete Comput. Geom. 9 (1993) 351–370. [9] D.Z. Du, F.K. Hwang, A proof of the Gilbert–Pollak conjecture on the Steiner ratio, Algorithmica 7 (1992) 121–136. [10] A. Dvoretzky, Some results on convex bodies and Banach spaces, in: Proceedings of Symposium on Linear Spaces, Jerusalem, 1961, pp. 123–160. [11] P. Fermat, Abhandlungen uV ber Maxima und Minima, Oswalds Klassiker Exakt. Wiss. 238 (1934). [12] B. Gao, D.Z. Du, R.L. Graham, A tight lower bound for the Steiner ratio in Minkowski planes, Discrete Math. 142 (1995) 49–63. [13] M.R. Garey, R.L. Graham, D.S. Johnson, The complexity of computing Steiner minimal trees, SIAM J. Appl. Math. 32 (197) 835 –859. [14] M.R. Garey, D.S. Johnson, The rectilinear Steiner tree problem is NP-complete, SIAM J. Appl. Math. 32 (1977) 826–834. [15] C.F. Gau;, Briefwechsel Gau;-Schuhmacher, in: Werke Bd. X,1, GVottingen, 1917, pp. 459 – 468. [16] E.N. Gilbert, H.O. Pollak, Steiner minimal trees, SIAM J. Appl. Math. 16 (1968) 1–29. [17] F.K. Hwang, D.S. Richards, P. Winter, The Steiner tree problem, Ann. Discrete Math. 53 (1992). [18] A.O. Ivanov, A.A. Tuzhilin, Minimal Networks – The Steiner Problem and its Generalizations, CRC Press, Boca Raton, FL, 1994. [19] A.C. Thompson, Minkowski Geometry, Cambridge University Press, Cambridge, 1996.