Vertex degrees of Steiner Minimal Trees in $ell_p^d$ and other smooth Minkowski spaces

V ertex degrees of Steiner Minimal T rees in ℓ d p and other smo oth Mink o wsk i spaces K. J. Sw anep o el Departmen t of Mathematics and Applied Mathematics Univ ersit y of Pretoria 0002 Pretoria, South Africa e-mail: konrad@math.up .ac.za Abstract W e find u pp er b oun d s for the degrees of vertices and Steiner p oints in Steiner Minimal T rees in the d -d imensional Banac h spaces ℓ d p inde- p endent of d . This is in contra st to Minimal Sp anning T rees, where the maximum degree of vertices gro ws exp on entially in d (R obins and Salow e, 1995). Ou r upp er bound s follo w from characteriza tions of singularities of SMT’s du e to Lawlo r and Morgan (1994), which w e extend, and certain ℓ p -inequalities. W e derive a general up p er b ound of d + 1 for the degree of vertices of an SMT in an arb itrary smo oth d -d imensional Banac h space (i.e. Minko wski space); th e same upp er b ound for Steiner p oints ha v ing b een found by Lawlor and Morgan. W e obtain a second upp er b ound for the degrees of vertices in terms of 1-summing norms. 1 In tro duction Given a metric space ( X, ρ ) a nd a set S ⊆ X , a Minimal Sp anning T r e e (MST) of S is a tree T with vertex set V ( T ) = S a nd edg e set E ( T ) such that X { x,y }∈ E ( T ) ρ ( x, y ) is minimal among all trees on S . A Steiner Minimal T r e e (S MT) of S is a tree T with vertex set V ( T ) satis- fying S ⊆ V ( T ) ⊆ X such that X { x,y }∈ E ( T ) ρ ( x, y ) is minimal among all trees on S with vertex sets satisfying S ⊆ V ( T ) ⊆ X . The elements of S ar e vert ic es , and the elements o f V ( T ) \ S ar e S teiner p oints of the SMT. Estimates for the lar gest degrees o f MST’s and SMT’s hav e consequences for the complexities of algo rithms that find suc h trees . F or ex ample, it is known that an MST o n n p oints can b e calculated in po lynomial time [2 ], while calcula ting the SMT in the euclidean or rectilinear planes is NP-hard [7, 8]. Upp er b ounds 1 for the degre es of vertices and Steiner p oints are used to reduce the sea r ch space of known exp onential time a lgorithms. Distance functions other tha n euclidean or r e ctilinear are sometimes used. The ℓ p metrics have b een found useful; see [15]. W e consider genera l Min- ko wski spaces, i.e. finite dimensional Banach spaces, and then sp ecialize to ℓ d p , d -dimensional r eal linear s pace with norm   ( x 1 , . . . , x d )   p =  d X i =1 | x i | p  1 /p . It is known that in a Minko wski space, the largest deg ree of an MST is eq ua l to the so -called Hadwiger n um ber H ( B ) of the unit ba ll B o f the space [3]. F or each 1 ≤ p ≤ ∞ ther e is a n ex po nential low er bound for the Hadwiger num b er of ℓ d p , H ( B d p ) > (1 + ǫ p ) d [19]. In co nt rast to this, we sho w in Section 4 that the degrees of b o th vertices and Steiner p oints of an SMT in ℓ d p (1 < p < ∞ ) ar e bounded ab ove by functions of p a lo ne, indepe ndent of d . F or p > 2 w e derive a ge ner al upp er b ound of 7, with v arious sharp er v a lues for sp ecific p . F or 1 < p < 2 how ever, we find a n upper bo und ex p o nential in p ∗ := p/ ( p − 1 ), and a low er b ound linear in p ∗ , as p tends to 1 . Thus with resp ect to the SMT proble m, ℓ d p behaves very similarly to euclidean s pace, wher e b oth v ertices and Steiner p oints have degree at most 3. F or g eneral d -dimensional s mo oth Minkowski spaces, it is known that the degree of a Steiner p oint is at mos t d + 1 [14]. In Section 3 w e show tha t this upper b ound also holds for the degr ee of a vertex in a n SMT. The pro of has t wo ingredients. Firstly , in Section 2 w e derive a characterization of the lo ca l structure o f a vertex in a n SMT (Theorem 2) simila r to the c haracteriz a tion of Steiner p oints due to Lawlor and Morg an [14]. W e als o r ederive their character- ization, paying attention to so me combinatorial subtleties (Theor em 1). Both deriv ations are co mpletely ele mentary . The seco nd ingr edient is Theor em 4, which g eneralizes a result of [6 ] a nd [14], th us answering a question in [21]. In Theor e m 5 we also obtain an upp er b ound for the degr ees of vertices and Steiner p o ints in terms of the 1-s umming norm of the dual of the space. 2 Deriv ation of the singularit y c haracterizations Theorem 1 b elow, due to [14], provides a c haracteriza tion of the str ucture of the neig hbourho o d of a Steiner po int in an SMT in a smo o th Minko wski spa ce. W e g ive a similar characterization of the str ucture of the neighbourho od o f a vertex in an SMT in Theorem 2. Both characterizatio ns ar e in terms o f unit vectors in the dual of the Mink owski space. W e now r ecall some fa c ts ab out dual spaces. Note that the discussion b elow per tains to finite dimensio nal Bana ch spaces, i.e. Minko ws ki spaces; s ee [23]. F or any d -dimensional r eal vector space X , the dual of X , denoted by X ∗ , is the vector s pace of linear functionals x ∗ : X → R . This dual is also a d - dimensional vector s pa ce. W e denote application of x ∗ ∈ X ∗ to x ∈ X by h x ∗ , x i . If X is furthermore a Minkowski spac e with norm k·k , then k x ∗ k ∗ = sup k x k≤ 1 h x ∗ , x i defines a nor m on X ∗ . 2 W e say that a Minko wski space is smo oth if lim t → 0 k x + th k − k x k t =: f x ( h ) exists for all x, h ∈ X with x 6 = 0. It follows easily that f x ∈ X ∗ , k f x k ∗ = 1 and h f x , x i = k x k . A linear functional x ∗ ∈ X ∗ is a norming functional of x if x ∗ satisfies h x ∗ , x i = k x k and k x ∗ k ∗ = 1. Each no n-zero v ector in a Minko wski space has a no rming functional (the Hahn- Banach theor em). A Minko w s ki space is smo oth iff each non-zer o v ec to r has a unique nor ming functional. A Minkowski space X is strictly c onvex if k x k = k y k = 1 and x 6 = y imply that   1 2 ( x + y )   < 1 , equiv alently , that the bo undary of the unit ball of X do es not contain any straight line se gment. A Minkowski space X is smo oth [strictly conv ex] iff X ∗ is strictly con vex [smo oth]. The balancing a nd collapsing co nditions in Theo rems 1 and 2 thus o ccur in a strictly conv ex space. W e say that a finite s et of unit vectors x 1 , . . . , x m ∈ X satisfies the b alancing c ondition if m X i =1 x i = 0 , (1) and satisfies the c ol lapsing c ondition if    X i ∈ J x i    ≤ 1 for each J ⊆ { 1 , . . . , m } . (2) Note that the ab ov e balancing condition is the character iz a tion of the so-ca lled F ermat point of a se t of p oints in a smo o th Minkowski spac e in the non-a bsorbing case (i.e. where the F e r mat p oint differs from the given p oints) in terms of norming functiona ls , derived in [1 ]. Theorem 1 (Lawlor and Morgan [1 4]) . L et a 1 , . . . , a m b e distinct non-zer o p oints in a smo oth Minkowski sp ac e X . F or e ach i = 1 , . . . , m , let a ∗ i b e the norming functional of a i . Then the tr e e c onn e cting e ach a i to 0 is an SMT of S = { a 1 , . . . , a m } iff { a ∗ 1 , . . . , a ∗ m } satisfies the b alancing and c ol lapsing c ondi- tions in X ∗ . Pr o of. ⇒ : Since we hav e an SMT, for any x ∈ X m X i =1 k a i − x k ≥ m X i =1 k a i k , i.e. for any unit vector e ∈ X the function φ e ( t ) := m X i =1 ( k a i + te k − k a i k ) ≥ 0 attains a minimum at t = 0. F or s ufficiently sma ll t , a i + te 6 = 0, and φ e ( t ) is differentiable at 0, with φ ′ e (0) = 0. But φ ′ e (0) = lim t → 0 m X i =1 k a i + te k − k a i k t = m X i =1 h a ∗ i , e i . 3 Therefore, P m i =1 a ∗ i = 0. Secondly , given J ⊆ { 1 , . . . , m } , define a tree T J as follows: Connec t { a i : i ∈ J } to an arbitrary p oint x , co nnect { a i : i / ∈ J } to 0, and connect x to 0. Then the total length of T J is not smaller than P m i =1 k a i k : X i ∈ J k a i − x k + X i / ∈ J k a i k + k x k ≥ m X i =1 k a i k , i.e. for any unit vector e the function ψ e ( t ) := X i ∈ J ( k a i − te k − k a i k ) + | t | ≥ 0 attains a minimum at t = 0. Ho wev er, ψ e is not differentiable at 0. Circum- ven ting this difficult y , we calculate 0 ≤ lim t → 0 + ψ e ( t ) t = lim t → 0 + X i ∈ J k a i − te k − k a i k t + 1 = X i ∈ J h a ∗ i , − e i + 1 and  P i ∈ J a ∗ i , e  ≤ 1 fo r all unit e . Thus   P i ∈ J a ∗ i   ∗ ≤ 1. ⇐ : Let a ∗ 1 , . . . , a ∗ m ∈ X ∗ satisfy (1) a nd (2), and let T b e any SMT of { a 1 , . . . , a m } . W e hav e to show that X { x,y }∈ E ( T ) k x − y k ≥ m X i =1 k a i k . F or i ≥ 2, let P i be any no n- ov erla pping path in T fro m a 1 to a i , i.e. P i = x ( i ) 1 x ( i ) 2 . . . x ( i ) k i with x ( i ) 1 = a 1 , x ( i ) k i = a i and { x ( i ) j , x ( i ) j +1 } distinct edges in E ( T ) for j = 1 , . . . , k i − 1 . Note that each edge of T is used in some P i , since the union of the paths is a connected subgr aph of T . F or each edg e e ∈ E ( T ) we as s ign a directio n dep ending on the wa y e is traversed in some P i containing e . This direction is unambigious, since if t wo pa ths w ould give conflicting directions, their union would contain a cycle. W e denote a dir ected edge from x to y b y ( x, y ) = ~ e a nd the set of directed edg es by ~ E ( T ). F o r eac h ~ e ∈ ~ E ( T ), let 4 S ~ e := { i ≥ 2 : ~ e ∈ P i } . Then m X i =1 k a i k = m X i =1 h a ∗ i , a i i = m X i =2 h a ∗ i , a i − a 1 i (b y the balancing condition) = m X i =2 k i − 1 X j =2  a ∗ i , x ( i ) j +1 − x ( i ) j  = X ~ e =( x,y ) ∈ ~ E ( T ) X i ∈ S ~ e h a ∗ i , y − x i ≤ X ~ e =( x,y ) ∈ ~ E ( T )    X i ∈ S ~ e a ∗ i    ∗ k x − y k ≤ X ( x,y ) ∈ ~ E ( T ) k x − y k (b y the co llapsing condition) . As mentioned in [1 4], the balancing a nd collapsing co nditions are still suffi- cient fo r the tree in the ab ove theorem to b e a n SMT in non-smo oth spaces, if (1) and (2) holds for some norming functional a ∗ i for each a i . A simila r remark holds for the next theorem. Theorem 2. Given p oints a 1 , . . . , a m 6 = 0 in a smo oth Minkowski sp ac e X , let a ∗ i b e the norming functional of a i . Then t he tr e e c onne cting e ach a i to 0 is an SMT of S = { 0 , a 1 , . . . , a m } iff { a ∗ 1 , . . . , a ∗ m } satisfies the c ol lapsing c ondition in X ∗ . Pr o of. Similar to the pro o f of the pr evious theor em. Note that there is no balancing condition, since we canno t per turb 0, a s 0 is in this case a vertex of the SMT. 3 Upp er b oun ds for smo oth M ink o wski spaces F or a Minko wski space X , let v ( X ) b e the la rgest degr ee of a vertex of an SMT in X , a nd s ( X ) the larges t deg ree o f a Steiner p oint in an SMT. In [14] it is shown that s ( X ) ≤ d + 1 if X is smo o th and d -dimensional. This inequality is sharp in the sense that there are spa ces and SMT’s where the degree o f d + 1 is attained. W e give a similar b ound for v ( X ): Theorem 3. F or a smo oth Minkowski sp ac e X of dimension d ≥ 2 , 3 ≤ s ( X ) ≤ v ( X ) ≤ d + 1 . The outer ine qualities ar e sharp in gener al. Pr o of. Theore ms 1 a nd 2 immediately imply s ( X ) ≤ v ( X ). In any 2-dimensional subspace o f the dua l X ∗ we can find t wo unit vectors x ∗ , y ∗ such that k x ∗ − y ∗ k ∗ = 1. Then the s e t { x ∗ , − y ∗ , y ∗ − x ∗ } satis fie s (1) and (2). 5 The euclidean spaces X = ℓ d 2 are ex a mples where s ( X ) = v ( X ) = 3. The rest of the theorem now follows from Theor em 2 and Theo rem 4 b elow. An exa mple where v ( X ) = d + 1 may b e constructed in the same way as for s ( X ), as is done in [14, Lemma 4.3]. The following theorem, s ug gested in [21], shar p ens results from [6] a nd [14] by eliminating the ba lancing condition from the h y p o theses. Theorem 4 . L et X b e a strictly c onvex d -dimensional Minkowski sp ac e. If x 1 , . . . , x m ∈ X ar e unit ve ctors satisfying the c ol lapsing c ondition, then m ≤ d + 1 . F urthermor e, if the b alancing c ondition is not satisfie d, i.e. P m i =1 x i 6 = 0 , then m ≤ d . Pr o of. Let x ∗ i ∈ X ∗ be norming functionals of x i . Firstly , for i 6 = j w e hav e 1 + h x ∗ i , x j i = h x ∗ i , x i + x j i ≤ k x i + x j k ≤ 1 by the collapsing co ndition, and thus h x ∗ i , x j i ≤ 0 for i 6 = j. Secondly , 0 ≤ D x ∗ i , − X j 6 = i x j E ≤     X j 6 = i x j     ≤ 1 . If  x ∗ i , − P j 6 = i x j  = 1, then x ∗ i is also a no rming functional of − P j 6 = i x j , which is now a unit vector. Then, since X is s trictly conv ex , it easily follows that x i = − P j 6 = i x j . Thu s, if P m i =1 x i 6 = 0 , then 0 ≤ D x ∗ i , − X j 6 = i x j E < 1 , and the diagona l of the matrix A =  h x ∗ i , x j i  m i,j =1 ma jorizes the rows. Thus A is inv er tible. Since A has ra nk at most d , w e obtain m ≤ d . If how ever P m i =1 x i = 0 , the ab ov e argument applied to x 1 , . . . , x m − 1 gives m − 1 ≤ d . Note that in the above pro of, we do no t nearly use the full force o f the collapsing condition. F or the next b ound, we re call a notio n from the lo cal theo ry o f Bana ch spaces. The absolutely summing c onstant or the 1 -summing norm (of the iden- tit y op era tor on) a Mink owski spa ce X is defined to be π 1 ( X ) := inf n c > 0 : ∀ x 1 , . . . , x m ∈ X : m X i =1 k x i k ≤ c max ǫ i = ± 1    m X i =1 ǫ i x i    o . This notion has b een studied ex tens ively; see e.g. [1 6, 5 , 20, 12, 9, 13]. Note that the quantit y (2 π 1 ( X )) − 1 has a lso b een called the Macphail c onstant in the literature. Theorem 5. F or a smo oth Minkowski sp ac e X , s ( X ) ≤ v ( X ) ≤ 2 π 1 ( X ∗ ) . 6 Pr o of. Let x ∗ 1 , . . . , x ∗ m ∈ X ∗ be unit vectors satisfying the colla psing co ndition, with m = v ( X ). Then, for an y seq uence o f sig ns ǫ i = ± 1 , i = 1 , . . . , m we hav e k P i ǫ i x ∗ i k ∗ ≤ 2 , hence m = m X i =1 k x ∗ i k ∗ ≥ m 2 max ǫ i = ± 1    m X i =1 ǫ i x ∗ i    ∗ , implying tha t m 2 ≤ π 1 ( X ∗ ). It is known that √ d ≤ π 1 ( X ) ≤ d for any d -dimensional X [12]. W e thus obtain an upp er b ound worse than that o f Theo r em 3, although it is of the sa me order. It is how ever p oss ible in principle to obtain bo unds b etter than that of Theorem 3 for sp ecific spaces. Howev er, we cannot do better than 2 √ d . 4 Upp er b oun ds for ℓ d p Restricting ourselves to the smo o th ca se 1 < p < ∞ , we recall that the dual of ℓ d p is ( ℓ d p ) ∗ = ℓ d p ∗ , wher e 1 /p + 1 /p ∗ = 1. W e use the Khinchin inequalities with the b est constants, due to [22] and [1 0, 11]. Khinc hi n’s inequaliti es. F or any 1 ≤ q < ∞ ther e exist c onstants A q , B q > 0 such that for any a 1 , . . . , a n ∈ R we have A q  n X i =1 a 2 n  1 / 2 ≤  2 − n X ǫ i = ± 1    n X i =1 ǫ i a i    q  1 /q ≤ B q  n X i =1 a 2 n  1 / 2 . F or q ≥ 2 we have A q = 1 , B q = √ 2  Γ( q +1 2 ) / √ π  1 /q , and for 1 ≤ q ≤ 2 , B q = 1 , A q = ( 2 1 / 2 − 1 /q if q < q 0 , √ 2  Γ( q +1 2 ) / √ π  1 /q if q ≥ q 0 , wher e q 0 ≈ 1 . 84 74 is define d by Γ  q 0 +1 2  = √ π 2 , 1 < q 0 < 2 . The following le mma is a nalogous to [4 , Hilfsatz 4 ]. W e omit the pro of, which e asily follows from calc ulus . Lemma 6. L et x, y ∈ R and 1 ≤ q ≤ 2 . Then | x + y | q ≥ 2 q − 2  | x | q/ 2 sgn x + | y | q/ 2 sgn y  2 . The earlies t reference w e could find to the following lemma is Rankin [17]. Lemma 7. L et x 1 , . . . , x m ∈ ℓ d 2 satisfy k x i k 2 = 1 and h x i , x j i < − 1 / n for i 6 = j , wher e n is a p ositive inte ger. Then m ≤ n . Pr o of. 0 ≤    m X i =1 x i    2 2 = m X i =1 k x i k 2 2 + 2 X i (log 3) / (log 2), then 2 1 − q − 1 < − 1 3 . By Lemma 7 we obtain m ≤ 3, and (3) follows. Similar ly , if p < (log 8 − log 3) / (log 4 − log 3), then 2 1 − q − 1 < − 1 4 , hence m ≤ 4, and (4) follows. F or the remaining estimates we apply K hinchin’s inequalities. W e may as- sume in the ligh t of (3) a nd (4 ) that p ≥ (log 8 − log 3 ) / (log 4 − lo g 3), i.e. q ≤ (lo g 8 − log 3) / (log 2) < q 0 . Th us A q = 2 1 / 2 − 1 /q . By (2) we hav e for any 8 sequence of signs ǫ i = ± 1 , i = 1 , . . . , m that k P m i =1 ǫ i x i k q ≤ 2. Therefore , 2 q ≥ d X n =1 2 − m X ǫ i = ± 1    m X i =1 ǫ i x i,n    q ≥ d X n =1 A q q  m X i =1 x 2 i,n  q/ 2 (Khinchin’s inequa lity) = A q q d X n =1    | x i,n | q  i   2 /q (where  | x i,n | q  m i =1 ∈ ℓ m 2 /q ) ≥ A q q      d X n =1 | x i,n | q  i     2 /q (triangle inequa lity in ℓ m 2 /q ) = A q q  m X i =1 k x i k 2 q  q/ 2 = A q q m q/ 2 , and m ≤ 4 / A 2 q = 2 3 − 2 /p < 8 . Estimates (5), (6) and (7) no w follow. Theorem 9. L et 1 < p < 2 and d ≥ 3 . Then min( d, f ( p ∗ )) ≤ s ( ℓ d p ) , v ( ℓ d p ) ≤ min( d + 1 , 2 p ∗ ) , (8) wher e for q > 2 , f ( q ) := max { d : 2( d − 2) q + ( d − 2)2 q ≤ ( d − 1) q + d − 1 } . In p articular, f ( q ) ≥ 3 for q > 2 , f ( q ) ≥ 4 for q ≥ 3 . 2106 7 , f ( q ) ≥ 5 for q ≥ 3 . 4009 3 , f ( q ) ≥ ⌈ q / log 2 ⌉ for q ≥ 3 . 6924 7 . Pr o of. Let q := p ∗ = p/ ( p − 1). The upp er bound follows from Theor em 3 a nd an application of Khinchin’s inequalities: 2 q ≥ d X n =1 2 − m X ǫ i = ± 1    m X i =1 ǫ i x i,n    q ≥ d X n =1  m X i =1 x 2 i,n  q/ 2 (Khinchin’s inequa lity) = d X n =1 k x i k q 2 ≥ d X n =1 k x i k q q = m (monotonicity o f q -norms) . F or the low er b ound w e may assume that d ≥ 4. Let x i be the vector in ℓ d q with d − 1 in its i ’th co ordina te, and − 1 in the remaining coo r dinates, 9 for i = 1 , . . . , d . Let ˆ x i := k x i k − 1 q x i . Then { ˆ x i : i = 1 , . . . , d } s atisfies the balancing conditio n (1). This set will a lso satisfy the collapsing condition iff fo r all 2 ≤ k ≤ d/ 2, g ( k , d, q ) := k ( d − k ) q + ( d − k ) k q ≤ ( d − 1) q + d − 1 = g (1 , d, q ) . By differentiating with resp ect to q and using 2 ≤ k ≤ d/ 2, it is ea sily seen that if g ( k , d, q ) ≤ g (1 , d, q ) holds for some q = q ′ , then it will hold for a ll q ≥ q ′ . The following numerical facts are easily verified: g ( k , d, q ) ≤ g (2 , d, q ) for 4 ≤ d ≤ 7 , 2 ≤ k ≤ d/ 2 a nd p ≥ 3 . 2 , g (2 , 4 , q ) ≤ g (1 , 4 , q ) for q ≥ 3 . 2106 6 . . . , g (2 , 5 , q ) ≤ g (1 , 5 , q ) for q ≥ 3 . 4009 2 . . . , g (2 , 6 , q ) ≤ g (1 , 6 , q ) for q ≥ 3 . 6924 6 . . . , and g (2 , 7 , q ) ≤ g (1 , 7 , q ) for q ≥ 4 . 0934 5 . . . . It is now sufficien t to show for d ≥ 8 and q = ( d − 1) log 2 that g ( k, d, q ) ≤ g (2 , d, q ) ≤ g (1 , d, q ) for all 2 ≤ k ≤ d/ 2. Firstly , note that in this case g (2 , d, q ) ≤ g (1 , d, q ) is equiv alent to 2 1+( d − 1) log( d − 2) + ( d − 2)2 ( d − 1) log 2 ≤ 2 ( d − 1) log( d − 1) + d − 1 , which is easily verified for d ≥ 8. Secondly , to show that g ( k , d, q ) ≤ g (2 , d, q ) it is sufficient to show that f ( x ) := x (1 − x ) q + (1 − x ) x q , 2 d ≤ x ≤ 1 2 attains its maximum at x = 2 d . T o see this, it is in turn sufficient to show that f ′ ( x ) ≤ 0 for 2 /d ≤ x ≤ 1 / 2. By setting y = (1 − x ) /x we find that it is sufficient to show that for 1 ≤ y ≤ d/ 2 − 1, x − q f ′ ( x ) = y q − qy q − 1 − 1 + q y =: h ( y ) ≤ 0 . By calculating the firs t and second deriv atives of h ( y ) and recalling that q > 3, it is seen that h ( y ) do es not attain its maximum if 1 < y < d/ 2 − 1. Since h (1) = 0, w e only hav e to show that h ( d/ 2 − 1) ≤ 0, whic h easily follo ws from q ≥ 4 and d ≥ 8. Ac kno wledgemen t This pap er is par t of the author’s PhD thesis being written under supe rvision of P rof. W. L. F ouch ´ e at the University of Pretoria. References [1] G. D. Chakerian and M. A. Ghandeha r i, T he F er mat Pr oblem in Minko wsk i Spaces, Ge om. De dic ata 17 (19 85), 227 –238. [2] D. Cheriton a nd R. E. T arjan, Finding minim um spanning tr ees, SIAM J. Comput. 5 (1 976), 724 –742 . 10 [3] D. Cieslik, Kno tengrade k¨ urzester B¨ aume in endlichdimensionalen B a- nachr¨ aumen, R osto ck Math. Kol lo q. 39 (19 90), 89 –93. [4] J. G. v an der Corput and G. Sch aake, An wendung einer Blich feldtschen Beweismethode in der Geometrie der Zahlen, Acta Ari thmetic a 2 (1937), 152–1 60. [5] A. Dvoretzky and C . A. Rogers , Abso lute and unco nditional conv ergence in normed linear s pa ces, Pr o c. Nat. A c ad. Sci. U.S.A. 36 (1 950), 19 2–19 7. [6] Z. F ¨ uredi, J. C. La garia s and F. Mor gan, Sing ularities of minimal sur faces and netw orks and related extremal problems in Minko ws ki s pace, in Dis- cr ete and Computational Ge ometry , (J. E. Go o dma n, R. Pollack, a nd W. Steiger, eds.), DIMACS Ser . Discrete Math. Theoret. Comput. Sci., vol. 6, Amer. Math. Soc ., Providence, RI, 1991 , pp. 95 – 109. [7] M. R. Ga rey , R. L. Graham and D. S. Johnson, The complexity of com- puting Steiner minimal trees , SIAM J. Appl. Math. 32 (19 77), 835– 839. [8] M. R. Garey and D. S. Johnso n, The rectilinear Steiner tree pro blem is NP-complete, SIA M J. Appl. Math. 32 (1977), 82 6–83 4 . [9] D. J. H. Ga rling and Y. Gordo n, Relations b etw een some constants as- so ciated with finite dimensional Banach spaces, Isr ael J. Math. 9 (1 971), 346–3 61. [10] U. Haag erup, Les meilleurs cons tantes de l’in ´ egalit´ e de Khintc hine, C. R. A c ad Sci. Paris A286 (1978 ), 2 59–2 62. [11] U. Haager up, The bes t constants in the Khin tchine inequality, Studia Math. 70 (198 2), 231–2 83. [12] M. I. Kadets and M. G. Snobar, Certain functiona ls on the Minko wsk i compactum, Mat. Zametki 10 (19 71), 453–45 7 (Russian). English tra nsl. Math. Notes 10 (1971), 69 4–69 6. [13] H. K¨ onig and N. T omczak - Jaeger mann, Bounds for pro jection constan ts and 1-summing norms, T r ans. Amer. Math. S o c. 320 (19 90), 799 – 823. [14] G. R. Lawlor a nd F. Morgan, Paired ca librations applied to so ap films, im- miscible fluids, and surfaces and netw orks minimizing other norms, Pacific J. Math. 166 (19 9 4), 55 –82. [15] R. F. Lov e and J . G. Mor r is, Mo delling inter-city r oad distances by math- ematical functions, J. Op er. R es. S o c. 23 (1972), 61–7 1. [16] M. S. Macphail, Absolute and unconditional co nv erg e nce, Bul l. Amer. Math. So c. 53 (1 9 47), 121 –123 . [17] R. A. Rankin, On the clos est pa cking of spheres in n dimensions, A nn. of Math. 48 (19 47), 1062 –108 1. [18] R. A. Rankin, On sums of p ow ers o f linea r forms I, I I, A nn. of Math. 50 (1949), 691 –704 . 11 [19] G. Ro bins and J. S. Salow e, Low-degree minimum spanning trees , Discr ete Comput. Ge om. 14 (1995), 151– 165. [20] D. Rutovitz, Some par ameters asso ciated with finite dimensional Banach spaces, J. L ondon Math. So c. 40 (1 965), 241 –255 . [21] K. J . Swanepo el, Ex tr emal problems in Minkowski spa ce related to minimal net works, Pr o c. Amer. Math. So c. 124 (1996), 2 5 13–2 518. [22] S. J. Szarek, O n the best consta nt s in the K hintc hine inequalit y, Studia Math. 58 (19 76), 197– 208. [23] A. C. Thompson, Minko wski Geo metry , Encyclo p e dia of Mathematics a nd its Applications 63, Cambridge Universit y Press, 199 6. 12