The number of generalized balanced lines

The n um b er o f gen era lized balanced lines ∗ Da vid Orden † P edro Ramos ‡ Gelasio Salazar § Abstract Let S be a set of r red p oin ts and b = r + 2 δ blue p oints in general position in the plane, with δ ≥ 0. A line ℓ de ter mined by them is b alanc e d if in each op en half-plane bo unded by ℓ the difference betw een the num b er of blue p oin ts a nd red points is δ . W e show that every s et S a s a bov e h a s a t le a st r balanced lines . The main techniques in the pro of are rotations a nd a generalization, sliding rotations , introduced here. 1 In tro duction Let B and R b e, resp ectiv ely , sets of b lue and red p oin ts in the p la n e, and let S = B ∪ R b e in general p osition. Let r = | R | an d b = | B | = r + 2 δ , with δ ≥ 0. F urthermore, w e are give n w eight s ω ( p ) = +1 for p ∈ B and ω ( q ) = − 1 f o r q ∈ R . Giv en a halfp la n e H , its weig ht is then defined as ω ( H ) = P s ∈ S ∩ H ω ( s ). Here and throu gh ou t this pap er, halfplanes are op en unless otherwise stated. Definition 1. A line ℓ determined by t wo p oin ts of S is b alanc e d if th e tw o halfplanes it defines ha ve weig ht δ . Ob serv e that this implies that the t wo p oin ts of S spanning ℓ hav e differen t colors. F or δ = 0, we obtain the original result, as conjectured by George Baloglo u , and prov ed b y P ac h and Pinc h asi via circular sequ e n ce s: Theorem 1 ([3]) . L et | R | = | B | = n . Every set S as ab ove determines at le ast n b alanc e d lines. This b ound is tight. Tigh tness is sh o w n, e.g., b y placing S on a conv ex 2 n -gon in suc h a w ay that R is separated from B by a str aight line. The general result w as prov ed by Sh a r ir and W elzl in an indirect m an n er, via an equiv- alence with a ve r y sp ecial case of the Generalized Low er Bound T heo r em. This motiv ated them to ask for a more direct and simp le r pro of. ∗ This w ork sta rt ed at the 6th Ib eri an W orkshop on Computational Geometry , in Aveiro, and w as concluded while Gelasio Salazar was v i siting Departament of Mathematics of Alcal´ a U n iv ersity under the program Giner de los R ´ ıos. † Departamento de Matem´ aticas, Universidad de Alcal´ a, david.orden @uah.es , partially supp orted by gran t MTM2008 - 0 4699-C03-02. ‡ Departamento de Matem´ aticas, Universidad de Alcal´ a, pedro.ramos @uah.es , partially supp orted by gran t MTM2008 - 0 4699-C03-02. § Instituto de F ´ ısica, U niv ersidad Aut´ onoma de San Luis Po t os ´ ı, Mexico, gsalazar@dec1.ifis ica.uaslp.mx 1 Theorem 2 ([4]) . L et B and R b e, r esp e ctively, sets of blue and r e d p oints in the plane, and let S = B ∪ R b e in gener al p osition. L et r = | R | and b = | B | = r + 2 δ , with δ ≥ 0 . The numb er of lines that p ass thr ough a p oint in B and a p oint in R , and such that the two induc e d halfplan e s have weight δ is at le ast r . This numb er is attaine d if R and B c an b e sep ar ate d by a line. In this pap er we giv e a simple pro o f of Theorem 2 using elemen tary geometric techniques. Therefore, via the r esults in [4 ], w e pro vid e a geometric pr oof of the follo wing ve r y sp ecial case of the Generalized Lo w er Bound Theorem: Let P b e a con ve x p olyto p e which is the intersectio n of d + 4 halfspaces in general p osition in R d . Let its ed g es b e oriented according to a generic linear function (edges are directed from smaller to larger v alue; “generic” means that the fu n ct ion ev aluates to d istin ct v alues at the v ertices of P ). Theorem 3 ([4]) . The numb er of vertic es with ⌈ d 2 ⌉ − 1 outgoing e dges is at most the numb er of vertic es with ⌈ d 2 ⌉ outgoing e dges. All p roofs in this pap er can b e easily translated to the more general setting of circular sequences (see [2]). 2 Geometric tools W e assum e that co ordinate axes a r e c hosen in such a w a y that all p oint s ha ve different abscissa. The to ols w e u se are inspired in the rotational mov emen t in tro duced by Erd˝ os et al. [1]. Definition 2. Let P ⊆ S . A P k -r otatio n is a family of directed lines P k t , where t ∈ [0 , 2 π ] is the angle measured from the v ertical axis, defined as follo ws: P k 0 con tains a single p oi nt of P , and as t increases, it rotates coun terclo c kwise in such a wa y that (i) | P ∩ P k t | = 1 except for a finite num b er of ev ents, w h en | P ∩ P k t | = 2; and (ii) whenev er | P ∩ P k t | = 1, there are exactly k p oin ts of P to th e righ t of P k t . The common p oin t P ∩ P k t = { p } is called the pivot , and it c h a n ge s p recise ly when | P ∩ P k t | = 2. Observe that P k 0 = P k 2 π . Definition 3. Let ℓ + and ℓ − denote, resp ectiv ely , the halfplanes to the righ t and to the left of ℓ . Let ω ( ℓ ) b e the weig ht of ℓ + . Give n a P k -rotation, we say that P k ≥ δ if ω ( P k t ) ≥ δ for ev ery t ∈ [0 , 2 π ], and similarly for the rest of inequalities. A rotation B k is δ -pr eserving if either B k ≥ δ or B k < δ . Symmetrically , R k is δ -pr eserving if either R k ≤ δ or R k > δ . Lemma 4. In an R k -r otatio n, tr ansitions δ δ + 1 and δ + 1 δ in ω ( R k t ) ar e always thr ough a b alanc e d line. In a B k -r otatio n, tr ansitions δ δ − 1 and δ − 1 δ in ω ( B k t ) ar e always thr ough a b alanc e d line. Pr o of. When a r e d p oin t is foun d during an R k -rotation, the weig ht of th e halfplane is pr e - serv ed b ecause the pivo t p oin t changes. Therefore, the c h a n g e δ δ + 1 happ ens when a blue p oin t is found in the head of R k t (Figure 1, left), wh ile δ + 1 δ happ ens wh e n a blue p oin t is found in the tail of R k t (Figure 1, right ). In b oth cases, the p oin ts define a balanced line. F or a B k -rotation, the pro of is identic al. 2 δ δ + 1 b r r b δ + 1 δ R k t R k t Figure 1: T ransitions in an R k -rotation are alw a ys th rough a balanced line. Claim 8.1 in [3] has now a more dir e ct pro of: Lemma 5. If r is o dd, ther e exists a b alanc e d line which is a halving line of S . Pr o of. Let k = ⌊ r 2 ⌋ and consider an R k -rotation. If R k 0 ≤ δ , then R k π > δ , and con versely . Therefore, there exist trans itions δ δ + 1 and δ + 1 δ in ω ( R k t ) wh ich, fr om Lemma 4 are alw a ys through a b a lanced lin e . Observ e that b oth trans it ions are through the same balanced line, with angles t 0 and t 0 + π . Remark 1. Let us observe that T heorem 1.4 in [3], whic h states that T heorem 1 is true when R an d B are separated by a line ℓ , has now an easier pro of: if w e start R k -rotations with a line parallel to ℓ , for eac h k there exist exactly one tr an s it ion δ δ + 1 and one transition δ + 1 δ w h ic h , from Lemma 4, corresp ond alw a ys to a balanced line. If r is ev en, there are 2 balanced lines for k = 0 , . . . , r 2 − 1, for a total of r balanced lines, while if r is o dd there are 2 balanced lines for k = 0 , . . . , ⌊ r 2 ⌋ − 1 and 1 balanced line for k = ⌊ r 2 ⌋ . Remark 2. L emm a s 4 and 5 conclude the p roof of Th eo r e m 1 if no R k -rotation is δ -preserving or if n o B k -rotation (with k ≥ δ ) is δ -preserving. Hence, in the follo w ing w e assume that there exists either at least one R k -rotation or one B k -rotation (with k ≥ δ ) wh ic h is δ -p reserving. Lemma 6. L et 0 ≤ j ≤ ⌊ r 2 ⌋ . If R j > δ then B j + δ ≥ δ , while if B j + δ < δ then R j ≤ δ . Pr o of. Consider the line R j t 0 . The halfplane ( R j t 0 ) + con tains j red p oin ts and b > j + δ blu e p oin ts. Therefore, the line B j + δ t 0 is to the righ t of ( R j t 0 ) + and con tains at most j r e d p oin ts. Then, ω ( B j + δ t 0 ) ≥ δ . The pr o of of the second statemen t is analogous. The next defin iti on generalizes the concept of P k -rotation in tw o different w a y s : parallel mo ve m ents are p ermitted and the n u m b er of p oin ts to the r ig ht of the line can c hange. Definition 4. A P -sliding r otat i o n consists in m oving a directed line ℓ con tinuously , starting with an ℓ 0 whic h con tains a single p oint p 0 ∈ P , and comp osing rotati on aroun d a p oi nt of P (the pivot) and parallel displacemen t (in either direction) until the next p oin t of P is found . F ur thermore, after a 2 π rotation is completed, the line ℓ 0 m us t b e reac hed again. This mov emen t is clearly a con tinuous curve in the sp ace of lines in th e plane. F or ins t an ce, if a line is parameterized as a p oin t in S 1 × R , a P -sliding rotation describ es a (non-strictly) angular-wise mon otone cur ve, with vertic al segmen ts corresp onding to parallel displacemen ts. Let Σ b e a P -sliding rotation. Let us denote by Σ t the line with angle t w ith resp ect to the ve r ti cal axis d e fi ned as follo ws: if there is no p aral lel displacemen t at angle t , then Σ t denotes the corresp onding line. O therwise, it den o tes the leftmost line corresp onding to angle t . 3 Definition 5. A P -sliding rotation Σ is p ositively oriente d if Σ t + π is to the left of Σ t for all t ∈ [0 , π ). That Σ ≥ δ , as w ell a s the rest of in equ a lities, is defined exactly as in Definition 3. Similarly , a B -sliding rotation Σ is δ -preserving if Σ ≥ δ , while an R -slidin g rotat ion is δ -preserving if Σ ≤ δ . The follo wing definition is the crux of the rest of th e pap er. Definition 6. Let S b e the set of all p ositiv ely orien ted, δ -preservin g B -slidin g rotations and R -sliding r o tations. The waist of a P -slidin g rota tion Σ ∈ S is min t ∈ [0 ,π ] | P ∩ Σ − t ∩ Σ − t + π | . W e denote by Γ the sliding r o tation of S with the smallest w aist. Note that the set S is non-empt y b ecause w e h a ve assum ed that there exist δ -preserving B k - or R k -rotations, wh ic h are a particular t yp e of s lid i n g rotations. F ur thermore, the wai s t tak es only a finite num b er of v alues, so it h a s a minimum. If th e minim um is not unique, we can p ick an y of the sliding rotations ac hieving it. 3 Main result Assume that Γ is a δ -preserving R -sliding rotatio n (i.e. Γ ≤ δ ). In this case, we will manage to pr ov e that there exist at least r b alanced lines. F or the case of Γ b eing a δ -p reserving B -sliding rotation, the same argumen ts w ould show that there exist at lea st b balanced lines. Lemma 7. L et Γ 0 and Γ π b e the lines achieving the waist of Γ , let Γ + 0 b e the close d halfplane to the right of Γ 0 and let F = R ∩ Γ + 0 . F or eve r y k ∈ { 0 , . . . , | F | − 1 } , during an F k -r otatio n a b alanc e d line is f o und. Similarly, let H = R ∩ Γ + π . F or e very k ∈ { 0 , . . . , | H | − 1 } , during an H k -r otatio n a b alanc e d line is found. Pr o of. Figure 2 illustrates the situation. On the one hand, F k 0 is to the righ t of Γ 0 and, since Γ is p ositiv ely orien ted, F k π is to the left of Γ π . T his imp li es that ther e is a t 1 ∈ [0 , π ] suc h th a t F k t 1 = Γ t 1 and therefore ω ( F k t 1 ) ≤ δ . On the other hand , F k 0 is to th e left of Γ π and F k π is to the right of Γ 0 , ther efore, th er e exists a t 2 ∈ [0 , π ] suc h that F k t 2 and Γ t 2 + π are the same line w it h opp osite directions. Since ω (Γ t 2 + π ) ≤ δ , then ω ( F k t 2 ) ≥ δ . If ω (Γ t 2 + π ) = δ and the line con tains a b lue p oin t, then it is a balanced lin e found in a transition δ δ + 1. Otherwise, ω ( F k t 2 ) > δ and hence a transition δ δ + 1 has o ccurred for a t ∈ ( t 1 , t 2 ). No w, observ e that R r F ⊂ Γ 0 − . Hence, in the F k -rotation for t ∈ [0 , π ], all the p oin ts in R r F are found b y th e head of th e line. This imp lies th a t a change δ δ + 1 in the weigh t of the righ t halfplane can only o cc u r when a blue p oin t is found in the head of the ra y (as in Figure 1, left), h e n ce defining a balanced line. The p roof for H is identica l. Before mo vin g on, let us p oint out that the | F | + | H | balanced lines giv en b y Lemma 7 are differen t, b ecause they ha ve exactly k p oin ts of F , resp ecti vely H , to the right. Let no w C Γ t b e the c entr al r e gion defin ed by the sliding rotation Γ at instan t t , defi ned as C Γ t = Γ − t ∩ Γ − t + π . Observe that, for the corr esp ondin g t , th e transitions δ δ + 1 in the pro of of Lemma 7 corresp ond to balanced lines ins id e or in the b oundary of the central region. 4 F k 0 F Γ 0 Γ π F k π Figure 2: Illustration of the pro of of Lemma 7. Lemma 8. L et G = R r ( F ∪ H ) . F or k ∈ { 0 , . . . , ⌈| G | / 2 ⌉ − 1 } , every G k -r otatio n has at le ast two tr ansitions b etwe en δ and δ + 1 , which c orr esp ond to lines inside or in the b oundary of the c entr al r e gi o n., i.e., for the c orr esp onding t , G k t ∈ C Γ t . Pr o of. Let us consider first the case when r is o dd and k = ⌊| G | / 2 ⌋ . G k 0 and G k π are the same line with opp osite d irect ions. Therefore, if ω ( G k 0 ) ≤ δ then ω ( G k π ) > δ and there m ust b e at least t wo transitions as stated. T hese transitions corresp ond to lines in the cen tral region b ecause Γ is p ositiv ely orien ted. F or the rest of cases, observe that, b y construction, G k 0 ∈ C Γ 0 . According to the v alue of ω ( G k 0 ), we distinguish t w o cases: • ω ( G k 0 ) ≤ δ . If th er e exist some v alues for wh ic h G k t = Γ t , let t 1 and t 2 b e, resp ectiv ely , the minimum and maxim um of them. If there is no suc h v alue, tak e t 1 = t 2 = 2 π . If G k tak es the v alue δ + 1 in the int erv al (0 , t 1 ) it m ust hav e transitions δ δ + 1 and δ + 1 δ , and the same is true for ( t 2 , 2 π ). Finally , observe th a t G k m us t tak e th e v alue δ + 1 at least on ce, b ecause in other case the sliding r o tation obtained by concatenating G k in (0 , t 1 ), Γ in ( t 1 , t 2 ) and G k in ( t 2 , 2 π ) would b e a δ -preserving sliding rotatio n of w aist smaller than the waist of Γ. • ω ( G k 0 ) > δ . If th er e exist some v alues for wh ic h G k t = Γ t , let t 1 and t 2 b e, resp ectiv ely , the minimum and maxim um of them. G k t tak es the v alue δ in the int erv als (0 , t 1 ) and ( t 2 , 2 π ) and therefore the lemma follo w s . In other case, if G k t tak es the v alue δ in the cen tral region, it must ha ve also transition δ δ + 1. Finally , if ω ( G k t ) > δ for all t ∈ [0 , 2 π ] we could construct a sliding rotation Σ con tradicting the choic e of Γ: for eac h t , consider as Σ t the parallel to G k t whic h passes through the fir st blu e p oin t to the righ t of G k t . It is easy to see that Σ t ≥ δ , b ecause b et w een Γ t and G k t there are alw a y s at least t wo blue p oints. The follo w in g lemma, wh ic h already app eared as Claim 6.4 in [3], w ill b e enough to conclude the pro of of Theorem 1. Lemma 9. T r ansitions δ δ + 1 and δ + 1 δ in a G k -r otatio n ar e always either a b alanc e d line or a δ + 1 δ tr ansition in an F j -r otatio n, j ∈ { 0 , . . . , | F | − 1 } or an H j -r otatio n, j ∈ { 0 , . . . , | H | − 1 } . Pr o of. On the one hand, a balanced line is ac hieved if there is su ch a transition b ecause a blue p oint is found . S e e Figure 1. On th e other h and, if the p oint in d ucing the transition is r ∈ R , th e n necessarily r ∈ R r G (since the G k -rotation changes piv ot w henev er a p oin t 5 of G is found). Figure 3 illustrates that a δ + 1 δ transition app ears for an F j -rotation with pivot g , b oth if f ∈ F is fou n d in the tail (left p ic tu r e) or if f ∈ F is found in th e head (righ t picture). Note that in the righ t p ict u re the weig ht of b oth halfplanes is δ + 1. The case δ δ + 1 in G k δ + 1 δ in G k δ + 1 δ in F j δ + 1 δ in F j g f g f Figure 3: T ransitions when a p oint f ∈ F ⊂ R foun d in a G k -rotation ind uces a δ + 1 δ transition in an F j -rotation. in wh ich the p oint found is h ∈ H works similarly . The follo wing simple observ ations show that the num b er of balanced lines is at least r whic h, together with Remark 1, finishes the pro of of Theorem 2: i) Lemma 7 give s | F | + | H | d iffe r en t balanced lin es. ii) Lemmas 8 and 9 giv e | G | lines which are, either a balanced line, or a δ + 1 δ transition at the cen tral region f or an F j - or H j -rotation. iii) Eac h transition in ii) forces a new δ δ + 1 transition at the cent r a l region for an F j - or H j -rotation wh ic h corresp ond, as in the p roof of Lemma 7, to a new balanced line. 4 Ac kno wledgemen ts The authors thank Jes ´ us Garc ´ ıa for helpful discu ssio n s. References [1] P . Erd˝ os, L. Lo v´ asz, A. S immons, E.G. S tr a u ss. Dissection graphs on planar p oin t sets. In A Su r v ey of Combinatorial The ory , North Holland, Ams t er d am, (19 73), 139–149. [2] D . Orden , P . Ramos, and G. Salazar, Balance d lines in tw o–colo u red p oin t sets. arXiv:0905 .3380 v1 [math.CO]. [3] J. Pac h and R. Pinc hasi. On the num b er of balanced lines, Discr ete and Computatio nal Ge ometry , 25 (2001 ), 611–6 28. [4] M. Sharir and E. W elzl. Balanced Lines, Halving T riangles, and the Generalized Lo wer Bound Theorem, In Disc r ete and Comp u ta tional Ge ometry — The Go o dman-Pol lack F estschrift , B. Arono v, S. Basu, J. P ach and M. S harir (Ed s.), Sp ringer-V erlag, Heidelb erg, 2003, p p. 78 9–798. 6