Balancing unit vectors

BALANCING UNIT VECTORS KONRAD J. S W ANEPOEL Abstra ct. Theorem A. Let x 1 , . . . , x 2 k +1 b e unit vectors in a normed plane. Then there exist signs ε 1 , . . . , ε 2 k +1 ∈ {± 1 } such that k P 2 k +1 i =1 ε i x i k ≤ 1. W e use the meth od of proof of th e ab ov e theorem to show the fol- lo wing p oint facil ity location result, generalizing Prop osition 6.4 of Y . S. Kupitz and H. Martini (1997). Theorem B. Let p 0 , p 1 , . . . , p n b e distinct p oints in a normed p lane such that for any 1 ≤ i < j ≤ n t h e closed angle ∠ p i p 0 p j conta ins a ra y opp osite some − − → p 0 p k , 1 ≤ k ≤ n . Then p 0 is a F ermat-T oricelli point of { p 0 , p 1 , . . . , p n } , i.e. x = p 0 minimizes P n i =0 k x − p i k . W e also prov e th e follo wing dynamic v ersion of Theorem A. Theorem C. Let x 1 , x 2 , . . . b e a sequence of un it ve ctors in a normed plane. Then there exist signs ε 1 , ε 2 , · · · ∈ {± 1 } such that k P 2 k i =1 ε i x i k ≤ 2 for all k ∈ N . Finally we discuss a v ariation of a tw o-pla yer balancing game of J. Sp encer (1977) related to Theorem C. 1. Introduction In this note we consider balancing results f or unit v ectors related to w ork of B´ ar´ an y and Grinb erg [1 ], Sp encer [6] and Peng and Y an [5] . W e apply these results to generalize a p oin t facilit y lo cation result from the Euclidean plane [4] to general normed planes. Finally we consider a dynamical balanc- ing prob lem for unit ve ctors in the form of a t w o-play er perfect information game. Our results will mainly b e in a normed plane X with norm k · k (except in Theorem 5, where higher-d im en sional normed spaces are also considered). 1.1. Balancing Unit V ectors. B´ a r´ an y and Grinberg [1] pro v ed the fol- lo wing: Theorem 1 ([1]) . L et x 1 , x 2 , . . . , x n b e a se quenc e of ve ctor s of norm ≤ 1 in a d -dimensio nal norme d sp ac e. Then ther e exist signs ε 1 , ε 2 , . . . , ε n ∈ {± 1 } such that k n X i =1 ε i x i k ≤ d. W e sharp en this theorem for an o d d num b er of unit ve ctors in a normed plane as follo ws. 1 2 KONRAD J. SW ANEPOEL Theorem 2. L et x 1 , . . . , x 2 k + 1 b e unit ve ctors in a nor me d plane. Then ther e exist signs ε 1 , . . . , ε 2 k + 1 ∈ {± 1 } such tha t k 2 k + 1 X i =1 ε i x i k ≤ 1 . This result i s b est p ossible in an y norm, as is seen by letting x 1 = x 2 = · · · = x 2 k + 1 b e an y unit vecto r. Th e pro of of this theorem is in Section 2. The metho d of pro of can also b e used to generalize a r esult on F erm at-T oricelli p oint s from the Eu clidean plane to an arbitrary normed plane (Section 1.3). B´ a r´ an y and Grinb erg also pro v ed the follo wing d ynamic b alancing theo- rem. Theorem 3 ([1 ]) . L e t x 1 , x 2 , . . . b e a se quenc e of ve ctors of norm ≤ 1 in a norm e d sp ac e. Then ther e exist signs ε 1 , ε 2 , · · · ∈ {± 1 } such that for al l k ∈ N , k k X i =1 ε i x i k ≤ 2 d. Again, for unit vec tors in a normed plane w e sh arp en this r esult as follo ws. Theorem 4. L et x 1 , x 2 , . . . b e a se quenc e of unit ve ctors in a norme d plane. Then ther e e xist signs ε 1 , ε 2 , · · · ∈ {± 1 } such that for al l k ∈ N , k 2 k X i =1 ε i x i k ≤ 2 . In the Euclide an plane the u pp e r b ound 2 c an b e r epla c e d by √ 2 . This result is b est p ossible in the rectilinear p lane w ith u nit ball a par- allelog ram — let x 2 i − 1 = e 1 and x 2 i = e 2 for all i ∈ N , wh ere e 1 and e 2 are an y ad j acen t vertic es of the unit b all. See S ection 3 for a pro of of this theorem. 1.2. Balancing Games. Th eorem 4 can b e used to analyze the f ollo win g v ariation of a t wo-pla ye r balancing game of Sp en cer. Fix k ∈ N a nd a normed space X . Let the starting p osition of the game b e p 0 = o ∈ X . In round i , Pla y er I chooses k unit v ectors x 1 , . . . , x k in X , and th en Pla yer I I c ho oses s igns ε 1 , . . . , ε k ∈ {± 1 } . Then the p osition is adjusted to p i := p i − 1 + P k j =1 ε j x j . Theorem 5. In th e ab ove game, Player II c an ke e p the se quenc e ( p i ) i ∈ N b ounde d iff X is at most two-dimensional and k is even. In fact, Player II c an for c e k p i k ≤ 2 for al l i ∈ N . The pro of is in Section 3. In [5] a v ector balancing game with a buffer is considered. Theorem 5 readily implies T heorem 4 of [5] in the sp ecial case of unit v ectors in a normed plane. BALANCING UNIT VECTORS 3 1.3. F ermat-T oricelli p oints. A p oin t p in a normed space X is a F ermat- T oric el li p oint o f x 1 , x 2 , . . . , x n ∈ X if x = p minimizes x 7→ P n i =1 k x i − x k . See [4] for a survey on the p roblem of findin g suc h p oint s. It is well -kno wn that in the Euclidean plane, if x 1 is in th e con vex h ull of n on-collinear { x 2 , x 3 , x 4 } , then x 1 is the (u nique) F er m at-T oricelli p oin t of x 1 , x 2 , x 3 , x 4 . Cieslik [2] generalized this result to an arbitrary norm ed plane (where the F ermat-T oricelli p oin t is not n ecessarily unique). There is also a generaliza- tion b y Kup itz and Martini [4, Prop osition 6.4] in another direction. Theorem. L et p 0 , p 1 , . . . , p 2 m +1 b e distinct p oints in the Euclide an plane such that for any distinct i and j the op en angle ∠ p i p 0 p j c ontains a r ay opp osite some − − → p 0 p k , 1 ≤ k ≤ 2 m + 1 . Then p 0 is the u nique F ermat-T oric el li p oint of { p 0 , p 1 , . . . , p n } . W e generalize this r esult as follo ws to an arb itrary normed plane. Theorem 6. L et p 0 , p 1 , . . . , p n b e distinct p oints in a norm e d plane such that for any distinct i and j the close d angle ∠ p i p 0 p j c ontains a r ay opp osite some − − → p 0 p k , 1 ≤ k ≤ n . Then p 0 is a F ermat-T oric el li p oint of { p 0 , p 1 , . . . , p n } . The pro of is in Section 2. Our seemingly w eak er hypotheses easily imp ly that n m ust b e o d d. The pro of in [4] of the E uclidean case uses rotations. Our pro of for any norm sho ws that it is r eally an affine resu lt. The correct affine to ol turns out to b e the f act that tw o-dimensional central ly symmetric p olytop es are zonotop es. 2. Zonogons A zonotop e P in a d -dimen sional vec tor sp ace X is a Mink o w ski sum of line segmen ts P = [ x 1 , y 1 ] + [ x 2 , y 2 ] + · · · + [ x n , y n ] where x 1 , . . . , x n , y 1 , . . . , y n ∈ X . It is w ell- kno wn that an y cen trally sym- metric tw o-dimensional p olytop e (or p olygon) is alw a ys a zonotop e (or zono- gon ) [8, Example 7.14]. In particular, if x 1 , . . . , x n are consecutive edges of a 2 n -gon P sy m metric around 0, then (1) P = n X i =1 [( x i +1 − x i ) / 2 , ( x i − x i +1 ) / 2] where w e tak e x n +1 = − x 1 . Lemma 7. L e t n ∈ N b e o dd and let P b e a p olygon with vertic es ± x 1 , . . . , ± x n with x 1 , . . . , x n in this or der on the b oundary of P . Then n X i =1 ( − 1) i x i = 1 2 n X i =1 ( − 1) i +1 ( x i +1 − x i ) ∈ P . Pr o of. The equatio n is simple to ve rify . That the righ t-hand side is in P follo w s from (1 ).  4 KONRAD J. SW ANEPOEL Note that Lemma 7 does n ot hold for even n . W e can now easily prov e Theorem 2. Pr o of of The or em 2. Fix a line through th e origin n ot cont aining any x i . Fix one of th e op en h alf planes H b ounded by this line. Then for eac h i , δ i x i ∈ H for some δ i ∈ {± 1 } . W e may ren um b er x 1 , . . . , x n suc h that δ 1 x 1 , . . . , δ n x n o ccur in this order on P = con v {± x i } . N o w tak e ε i = ( − 1) i δ i and apply Lemma 7, noting that P is conta ined in the un it ball.  Recall that the dual of a finite dimensional n ormed space X is the norm ed space of all lin ear fun ctionals on X with norm k φ k = max { φ ( u ) : k u k = 1 } . A norming functional φ of a non-zero x ∈ X is a linear f unctional satisfying k φ k = 1 and φ ( x ) = k x k . Re call that b y th e separation th eorem any n on- zero x ∈ X has a norming fu nctional (see e.g. [7 ]). The follo wing lemma is w ell-kno wn and easily pro v ed. See [4] for the Euclidean case and [3] for th e general case. W e only need the second case of the lemma, b u t w e also state the fi rst case for the sak e of completeness. Lemma 8. L et p 0 , p 1 , . . . , p n b e distinct p oints in a finite-dimensional norme d sp ac e X . (1) Then p 0 is a F ermat-T oric el li p oint of p 1 , . . . , p n iff p i − p 0 has a norming functional φ i (1 ≤ i ≤ n ) such that P n i =1 φ i = o , (2) and p 0 is a F ermat-T oric el li p oint of p 0 , p 1 , . . . , p n iff p i − p 0 has a norming functional φ i (1 ≤ i ≤ n ) such that k P n i =1 φ i k ≤ 1 . Pr o of of The or em 6. By Lemma 8 it is sufficien t to find n orm ing fu nctionals φ i of p i − p 0 suc h th at k P n i =1 φ i k ≤ 1. W e order p 1 , . . . , p n suc h that − − → p 0 p 1 , . . . , − − → p 0 p n are ordered counter-cl o c kwise. If p 0 ∈ [ p i , p j ] for some 1 ≤ i < j ≤ n , w e ma y choose φ i = − φ j . W e ma y therefore assume that p 0 / ∈ [ p i , p j ] for all distinct i, j . Thus for an y i , the op en angle ∠ p i p 0 p i +1 con tains a ray opp osite some − − → p 0 p k . W e n o w show that necessarily n is o dd and k ≡ i + ( n + 1) / 2 (mod n ). Since eac h o p en a ngle conta ins a t least one − p k , eac h op en angle con tai ns exac tly one suc h − p k , sa y − p k ( i ) . Th e line through p 0 and p k ( i ) cuts { p 1 , . . . , p n } in t w o op en half planes: On e half plane contai ns as many op en angles as p oints p i . Thus n is o dd, and k ( i ) ≡ i + ( n + 1) / 2 (mo d n ). It is no w p ossible to c ho ose normin g fu nctionals φ i of eac h p i − p 0 suc h that φ 1 , − φ m +1 , φ 2 , − φ m +2 , . . . a re consecutiv e vec tors on the unit circle in the du al normed plane. It is therefore sufficien t to prov e that in any norm ed plane, if we choose unit ve ctors x 1 , . . . , x n suc h that x 1 , . . . , x n , − x 1 , . . . , − x n are in this ord er on the unit circle, then k P n k =1 ( − 1) k x k k ≤ 1. This follo ws at once from Lemma 7 .  3. Online Balancing Pr o of of The or em 5. ⇒ W e assume that some inner pro duct structure has b een fixed on X . BALANCING UNIT VECTORS 5 If k is o dd then in round i Pla yer I c ho oses the k un it vect ors al l to b e the same unit v ecto r, orthog onal to p i − 1 . Then, indep enden t of the c hoice of signs by Pla yer I I, the Euclidean norm of p i gro ws > c √ i . If k is ev en and X is at least three-dimensional, Pla y er I fi nds u nit ve ctors e 1 and e 2 suc h that e 1 , e 2 , p i − 1 are mutually orthogonal, then in round i tak es e 1 for the fir st k − 1 u n it vecto rs, and e 2 for the last u nit v ect or. Again the Euclidean norm of p i will gro w > c √ i . ⇐ follo ws immediately from Lemmas 9 an d 10 b elo w.  Pr o of of The or em 4. follo ws immediatel y from the follo wing t w o lemmas.  Lemma 9. L et w , a, b b e ve ctors in a norme d plane such that k w k ≤ 2 , k a k = k b k = 1 . Then ther e exist signs δ , ε ∈ {± 1 } such that k w + δ a + εb k ≤ 2 . Pr o of. If a = ± b , then the lemma is trivial. So assume that a and b are linearly in d ep endent. Let w = λa + µb . Without loss of generalit y w e assume that λ, µ ≥ 0, and sho w that k w − a − b k ≤ 2. If λ = 0, then 0 ≤ µ ≤ 2 and k ( λ − 1) a + ( µ − 1) b k ≤ k a k + k ( µ − 1) b k ≤ 2. S o we may assu me th at λ > 0, and similarly , µ > 0. Then w e can wr ite a = − ( µ/λ ) b + (1 /λ ) w . T aking norms w e obtain 1 = k a k ≤ µ/λ + 2 /λ , and therefore, λ − µ ≤ 2. Similarly , µ − λ ≤ 2. So we already ha v e | ( λ − 1) − ( µ − 1) | ≤ 2. If fu rthermore λ + µ ≤ 4, we also obtain | ( λ − 1) + ( µ − 1) | ≤ 2, giving k ( λ − 1) a + ( µ − 1) b k ≤ | λ − 1 | + | µ − 1 | ≤ 2. In the remainin g case λ + µ ≥ 4 w e wr ite ( λ − 1) a + ( µ − 1) b as a non- negativ e linear com bination ( λ − 1) a + ( µ − 1) b = λ + µ − 4 λ + µ − 2 ( λa + µb ) + 2 + λ − µ λ + µ − 2 a + 2 − λ + µ λ + µ − 2 b, and apply the triangle inequalit y: k ( λ − 1) a + ( µ − 1) b k ≤ 2 λ + µ − 4 λ + µ − 2 + 2 + λ − µ λ + µ − 2 + 2 − λ + µ λ + µ − 2 = 2 .  Lemma 10. L et w, a, b b e ve ctors in the Euclide an plane such that k w k ≤ √ 2 , k a k = k b k = 1 . Then ther e exist signs δ, ε ∈ {± 1 } such that k w + δ a + εb k ≤ √ 2 . Pr o of. Note th at a + b ⊥ a − b . W rite p = a + b , q = a − b . Let m b e the midp oint of pq , and L the p er p endicular bisector of pq . Ass u me without loss that k p k ≥ k q k and that w is inside ∠ poq . W e no w sho w that k w − p k ≤ √ 2 or k w − q k ≤ √ 2. Note that as w v aries, min( k w − p k , k w − q k ) is maximize d on L . Let L and op in tersect in c (b et w ee n o and p ), and L and th e circle with cen tre o and rad iu s √ 2 in d (inside ∠ p oq ). S ee Figure 1. Then clearly max k w k≤ √ 2 min( k w − p k , k w − q k ) = m ax( k p − c k , k p − d k ) , 6 KONRAD J. SW ANEPOEL d c q m L p o Figure 1. and we hav e to sho w k p − c k ≤ √ 2 and k p − d k ≤ √ 2. Since k p k ≥ k q k , w e h a v e ∠ opq ≤ 45 ◦ and k p − c k = s ec ∠ opq ≤ √ 2. Since c is b et w een o and p , we ha v e ∠ omd ≥ 90 ◦ , h ence k m − d k 2 ≤ k d k 2 − k m k 2 = 2 − 1, and k p − d k 2 = k p − m k 2 + k m − d k 2 ≤ 1 + 1.  4. Conclud ing remark s It would b e in teresting to fin d higher dimensional generalizations of our results and metho d s. W e only mak e the follo wing remarks. P erhaps there is an analogue of Theorem 2 with an up p er b ound of d − 1 for n u n it v ecto rs in a d -dimensional normed s pace wh ere n 6≡ d (mo d 2). This w ould b e b est possib le, as the standard unit v ecto rs in the d -dimensional space with the L 1 norm sho w. Regarding Theorem 4, it is not even clear what the b est upp er b ound in Theorem 3 shou ld b e. B´ ar´ a n y and Grin b erg [1] claim that they can replace 2 d b y 2 d − 1. On the other hand , the up p er b ound cannot b e smaller than d , as the d -dimensional L 1 space sho ws [1]. As the negativ e part of Theorem 5 and th e r esu lts of [5] s ho w, an online metho d w ould hav e to ha v e a (suffi cien tly large) buffer where Pla yer I I can put v ec tors sup plied by Pla y er I and tak e them out in any order. W e finally remark that a naive generalizati on of Theorem 6 is not p ossible, ev en in Eu clidean 3-space . F or exa mple, using Lemma 8 it can b e sho wn that for a regular s im p lex with v ertices x i ( i = 1 , . . . , 4) there exists a p oint x 5 in the in terior of th e simplex such that x 5 is not a F ermat-T oricelli p oin t of { x 1 , . . . , x 5 } — we ma y tak e any x 5 sufficien tly near a vertex. ackno wledgments W e thank the referee for suggestions on imp ro ving the pap er. BALANCING UNIT VECTORS 7 Referen ces [1] I. B´ ar´ any and V. S. Grin b erg, On some c ombinatorial quest ions in finite-dimensional sp ac es , Linear A lgebra Appl. 41 (1981), 1–9. [2] D. Ciesl ik, Steiner minimal tr e es , N onconv ex optimization and its applications, vol . 23, Kluw er, D ordrec ht, 1998. [3] R. Durier and C. Mic h elot, Ge ometric al pr op erties of the F ermat-Web er pr oblem , Europ. J. O p er. Res. 20 (1985 ), 332–343. [4] Y. S. Kupitz and H . Martini, Ge ometr ic asp e cts of the gener alize d F ermat-T oric el li pr oblem , I ntuitiv e Geometry , Bolya i S o ciety Mathematical St u dies, v ol. 6, 1997 , pp. 55–127 . [5] H. P eng and C. H . Y an, Balancing game with a buffer , A dv. Ap pl. Math. 21 (1998), 193–204 . [6] J. Sp encer, Balancing games , J. Comb. Th. Ser B 23 (1977), 68– 74. [7] A. C. Thompson, Mi nkowski Ge ometry , Encyclopedia of Mathematics and its Ap pli- cations 63, Ca mbridge Universit y Press, 1996. [8] G. M. Ziegler, L e ct ur es on Polytop es , Graduate T ex t s in Mathematics 152, Springer- V erlag, New Y ork, 1995. Dep ar tment of Ma the ma tics and Applied Ma thema tics, Unive rsity of Pre- toria, Pretoria 0002, South Africa E-mail addr ess : konr ad@math.up .ac.za