Distance k-Sectors Exist

Distance k -sectors exist Keik o Imai 1 Akitoshi Ka w am ura 2 Ji ˇ r ´ ı Matou ˇ sek 3 Daniel Reem 4 T ak eshi T okuyama 5 Decem b er 2 0 09 Abstract The bise c tor of t wo nonempty sets P and Q in R d is the set of a ll p oin ts with equal distance to P and to Q . A distanc e k -se ctor of P and Q , where k ≥ 2 is an integer, is a ( k − 1)-tuple ( C 1 , C 2 , . . . , C k − 1 ) such that C i is the bise ctor of C i − 1 and C i +1 for every i = 1 , 2 , . . . , k − 1, where C 0 = P and C k = Q . This notion, for the case where P and Q are po in ts in R 2 , was introduced by Asano, Ma tou ˇ sek, and T okuyama, motiv ated by a question of Mur ata in VLSI design. They established the existence and uniqueness o f the distance tr isector in this sp ecial case. W e prove the existence of a distance k -sector for all k a nd for every t wo disjoint, nonempt y , clo sed sets P a nd Q in Euclidean spaces o f any (finite) dimension (uniqueness remains op en), or mo r e genera lly , in pro per geo desic spac e s. The core o f the pro of is a new no tion of k -gr adation for P and Q , whose existence (ev en in an arbitra ry metric space) is proved using the Kna ster–T arski fixed p oint theorem, b y a metho d introduced by Reem and Reich for a s lig h tly different purpo se. 1 In tro duction The bise ctor of tw o n onempt y sets X and Y in E uclidean space, or in an arb itrary metric space ( M , d ist), is defined as bisect( X, Y ) = { z ∈ M : dist( z , X ) = dist( z , Y ) } , (1) where dist( z , X ) = inf x ∈ X dist( z , x ) denotes the distance of z from a set X . Let k ≥ 2 b e an int eger and let P , Q b e disjoint nonempty sets in M called the sites . A distanc e k -se ctor (or simply k - se ctor ) of P and Q is a ( k − 1)-tuple ( C 1 , . . . , C k − 1 ) of nonempt y subsets of M such that C i = bisect( C i − 1 , C i +1 ) , i = 1 , . . . , k − 1 , (2) where C 0 = P and C k = Q (see Figures 1 and 2). Distance k -sectors we re in tro duced by Asano et al. [3], motiv ated b y a question of Murata from VLS I design: Su pp ose that w e are give n a top ology of a circuit la y er, and we need to put A preliminary form of the results w as annou nced in S ection 4 of [5]. F or th e case k = 3, some of the metho ds and results w ere found essen tially indep endently in [8 ]. 1 Department of Information an d S ystem Engineering, Chuo Un iv ersity . imai@ise.c huo-u.ac.jp 2 Department of Computer Science, U nive rsity of T oronto. kawamura@cs.toronto .edu 3 Department of App lied Mathematics, Charles Universit y . matou sek@kam.mff.cuni .cz 4 Department of Mathematics, The T echnion – Israel Institute of T ec hn ology . dr eam@tx.technion. ac.il 5 Graduate School of Information Sciences, T ohoku U nive rsity . tokuyama@da is.is.tohoku.ac. jp 1 P Q C 1 C 2 C 3 Figure 1: A 4-sector ( C 1 , C 2 , C 3 ) of sets P and Q in Euclidea n plane. Each p oint on the curve C i is a t the same distance from C i − 1 and C i +1 . Note that C 2 is not the bisector of P and Q . k − 1 wir es thr ough a corridor b etw een give n t w o sets of obstacles (mo dules and other wires) on the b oard. The circuit will ha v e a high failure p robabilit y if the gaps b et w een the wires are narro w. Whic h cu rv es should the wires follo w in order to minimize the failure p robabilit y? If k = 2, the cur ve should b e the distance bisector; in general, eac h curv e s hould b e the b isector of its adjacent pair of curv es, as stated in the definition of a k -sector. A similar problem o ccurs also in designing routes of k − 1 autonomous r ob ots mo ving in a narro w p olygonal corridor. Eac h r ob ot has its o wn p redetermined route (say , it is drawn on the flo or with a coloured tap e that the rob ot can recognize) and tr ies to follo w it. W e wan t to design the routes to b e far a w a y from eac h other so that the rob ots can easily a void collisio n. Despite its inn o cen t definition, it is nontrivial to find a k -sector ev en in Euclidean plane. The bisector of t wo p oint sites P and Q in R 2 is a line, and an elementa ry geometric argumen t sho ws that there is a distance 4-sector of them consisting of a straigh t line and t wo parab olas. Ho wev er, the problem w as not in v estigated for other v alues of k until Asano et al. [3] p ro v ed the existence and uniqueness of the 3-sector of t wo p oin ts in Euclidean plane. Chun et al. [4] extended this to the case where Q is a line segmen t. W e giv e the first p ro of of existence of d istance k -sectors in Euclidean spaces for a general k . This imp ro v es on the previous p ro ofs in generalit y and simplicit y ev en for k = 3. Main Theorem. Every two disjoint nonempty close d sets P and Q in Eu clide an sp ac e R d , or mor e gener al ly, in a pr op er ge o desic metric sp ac e, have at le ast one k -se ctor. Here, a metric space ( M , dist) is called pr op er if all closed b alls are compact. It is called ge o desic if for ev ery t w o distinct p oin ts x , y ∈ M there is a metric se gment in M connecting them, i.e., an isometric mapping γ : [ a, b ] → M of an in terv al [ a, b ] ⊂ R with γ ( a ) = x and γ ( b ) = y . In particular, a con ve x subs et of a norm ed space is a geo d esic metric space. Another example is the su rface of a sp h ere, wh ere th e distance b et w een tw o p oin ts is measur ed b y the length of the sh ortest path on the surf ace connecting them. Geo desic metric spaces are a r easonably general class of metric sp aces in w hic h our arguments go through, although one could p robably make up ev en more general conditions. Let us r emark that if dist( P , Q ) > 0 and k = 3, then the prop erness assumption can b e omitted; see [8] for a p ro of. 2 P Q Figure 2: A 7- sector of t wo sing le ton sets P and Q in Euclidean plane. On the other hand, k -sectors need not exist in arb itrary m etric spaces. A simple example for k = 3 is the subspace M = {− 1 , 0 , 1 } of the real line, P = { 1 } , and Q = {− 1 } . F rom no w on, unless otherwise noted, s ubscripts i and j r ange ov er 1, . . . , k − 1; for example, ( C i ) i stands for the k -tuple ( C 1 , . . . , C k − 1 ). Gradations One of the main steps in the pro of of the main theorem is int ro du cing the notion of a k - gr adation of P and Q , wh ich is r elate d to a k -sector but easier to w ork with. First, for nonempty sets X , Y in a metric space ( M , d ist ), w e defi ne th e dominanc e r e gion of X o ver Y by dom( X, Y ) = { z ∈ M : dist( z , X ) ≤ dist( z , Y ) } . (3) A k -gr adation b et we en nonempt y su bsets P and Q of M is a ( k − 1)-tuple ( R i , S i ) i of pairs of su bsets of M satisfying R i = dom( R i − 1 , S i +1 ) , S i = dom( S i +1 , R i − 1 ) , i = 1 , . . . , k − 1 , (4) where R 0 = P and S k = Q . Using the Knaster–T arski fixed p oin t theorem [9], w e p ro v e in Section 2 that k -gradations alw a ys exist: Prop osition 1. F or every nonempty sets P and Q i n an ar bitr ary metric sp ac e ( M , d ist ) , ther e exists at le ast one k -gr adation. The id ea of applying the Knaster–T arski theorem to a similar setting is from [7], w here it is us ed to p ro v e the existence of double zone diagr ams . A sligh t mo dification of Pr op osition 1 also h olds in the more general setting of m-sp ac es [7]. In Section 3, we establish the follo win g connection b et w een k -gradations and k -sectors. Prop osition 2. L et P , Q b e nonempty, disjoint, close d se ts in a pr op er ge o desic metric sp ac e. Then a ( k − 1) -tuple ( C i ) i of sets is a k -se ctor of P and Q if and only if C i = R i ∩ S i , i = 1 , . . . , k − 1 (5) for some k -gr adation ( R i , S i ) i b etwe en P and Q . 3 P C 1 C 2 Q Figure 3: A 3-sector ( C 1 , C 2 ) of P and Q under the ℓ 1 norm. F or instance, the k -secto rs ( C i ) i in Figures 1, 2 and 3 corresp ond to the k -gradations ( R i , S i ) i where eac h R i is th e un ion of C i and the region ab o ve it, and eac h S i is th e un ion of C i and the region b elo w it. The main theorem is an immediate consequence of Prop ositions 1 and 2 . 3-gradations and zone diagrams A zone diagr am of P , Q is, according to the general definition of Asano et al. [2], a pair of sets ( A, B ) such that A = dom( P , B ) and B = dom( Q, A ). By comparing the definitions, w e can see that if (( R 1 , S 1 ) , ( R 2 , S 2 )) is a 3- gradation for P , Q , then ( R 1 , S 2 ) is a zone diagram of P , Q . Con ve rsely , giv en a zone diagram ( A, B ), w e can set R 1 := A , S 2 := B , R 2 := dom( R 1 , Q ), S 1 := dom( S 2 , P ) to obtain a 3-gradation (w e note that R 2 and S 1 are un iquely determined by R 1 and S 2 ). The existence of zone diagrams of arbitrary tw o nonempty sets in an arbitrary metric s p ace (and ev en in the still more general setting of m-spaces) was pro ve d b y Reem and Reic h [7 , Theorem 5.6] . B y the ab o v e, it immediately implies the existence of 3-gradations, a sp ecial case of Pr op osition 1. Uniqueness Ka wa mura et al. [6] (also see [5] for a preliminary version) p ro v ed the existence and un iqueness of zone diagrams in R d (for fin itely man y closed and pairwise separated sites) under the Euclidean distance, and more generally , under an y smo oth and un iformly conv ex norm. By Prop osition 2 , this implies the un iqueness of trisectors und er the same conditions. This is the most general uniqueness r esult for k -sectors w e are a wa re of. F or general metrics, k -sectors need not b e unique. A s imple example, for the ℓ 1 metric in the plane (giv en by dist( x, y ) = | x 1 − y 1 | + | x 2 − y 2 | ), is sh o w n in Figure 3; essen tially , it was disco v ered by Asano and Kirkp atrick [1]. Th e set C 1 is a p olygonal curve, while C 2 is “fat”, consisting of t wo straigh t segmen ts and t wo qu adran ts. A different trisector is obtained as a mirror r eflectio n of th e one sho wn . Th us , uniqu eness of k -sectors requ ires some geometric assump tions on the un derlying metric space. W e will fur ther commen t on this issue in S ection 4. Construction of k -sectors Our existence pro of for k -secto rs, based on the Knaster–T arski theorem, is somewhat nonconstructive . In Section 4, w e discuss a more constructive appr oac h, whic h re-establishes Pr op osition 1 under stronger assu mptions, but whic h yields an iterativ e algorithm (in a s imilar spirit as in [2]). W e ha ve no rigorous r esults ab out the s p eed of its con v ergence, bu t in practice it h as b een used su ccessfully for app ro ximating k -sectors and dra wing pictures such as Figure 1. Su ch compu tations also su pp ort our b elief that k -sectors in Euclidean sp aces are uniqu e, at least for t wo p oin t sites in the plane. 4 2 The existe nce of k -gradations Here w e prov e Prop osition 1. A set L equipp ed with a p artial order ≤ is called a c omplete lattic e if every subs et D ⊆ L has an infimum V D (the greatest x ∈ L suc h that x ≤ y for all y ∈ D ) and a sup rem um W D (the least x ∈ L suc h that x ≥ y for all y ∈ D ). W e sa y that a function F : L → L on a complete lattice L is monoto ne if x ≤ y implies F ( x ) ≤ F ( y ). Knaster–T arski Theorem ([9]) . Every monotone function on a c omplete lattic e has a fixe d p oint. The pro of of th is theorem is simple: It is routine to v erify that the least and the greatest fixed p oin ts of a monotone fun ction F : L → L are giv en by ^ { x ∈ L : x ≥ F ( x ) } , _ { x ∈ L : x ≤ F ( x ) } , (6) resp ectiv ely . Pr o of of Pr op osition 1. Let L b e the set of all ( k − 1)-tuples ( R i , S i ) i of pairs of sub sets of the consid ered metric space M satisfying R i ⊇ P , S i ⊇ Q and R i ∪ S i = M . W e defin e the order ≤ on L by setting ( R i , S i ) i ≤ ( R ′ i , S ′ i ) i if R i ⊆ R ′ i and S i ⊇ S ′ i for all i = 1 , . . . , k − 1. It is easy to see that L with this order ≤ is a complete lattic e in whic h th e infimum and supremum of D ⊆ L are given b y ^ D =  \ ( R j ,S j ) j ∈D R i , [ ( R j ,S j ) j ∈D S i  i , _ D =  [ ( R j ,S j ) j ∈D R i , \ ( R j ,S j ) j ∈D S i  i . (7) W e define F : L → L b y F  ( R i , S i ) i  =  dom( R i − 1 , S i +1 ) , dom( S i +1 , R i − 1 )  i , (8) where R 0 = P and S k = Q . It is easy to see that F is wel l-defin ed and monotone. By the Knaster–T arski Theorem, F h as a fixed p oint , whic h is a k -gradation by definition. 3 Dominance regions, k -gradations, and k -sectors The goal of this section is to prov e Prop osition 2. W e write ∂ Z fo r the b oundary of a closed set Z . W e b egin with observing th at, for arbitrary nonempty sets X , Y in any metric space, the set bisect( X , Y ) = dom( X , Y ) ∩ dom( Y , X ) con tains ∂ dom( X , Y ). Moreo ver, if the metric s pace is geo desic (and hen ce connected), then bisect( X , Y ) is n onempt y . F or otherwise, dom( X, Y ) and dom( Y , X ) w ould b e t wo disjoin t closed sets co vering the wh ole space. Lemma 3. L et X , Y , Z b e nonempty close d sets in a pr op er ge o desic metric sp ac e. Note that D = dom( X, Y ) and C = b isect( X, Y ) ar e nonempty. If D and Z ar e disjoint, then (a) dom( D , Z ) = d om( C , Z ) , d om( Z, D ) = dom( Z, C ) , (b) bisect( D , Z ) = b isect( C, Z ) . Pr o of. P art (b) follo ws from (a) using bisect( X, Y ) = dom( X , Y ) ∩ dom( Y , X ). T o sho w (a), w e claim that dist( a, Z ) > dist( a, C ) for all a ∈ D . (9) 5 Indeed, let z ∈ Z b e a p oin t attaining the distance to a ; i.e., dist( a, z ) = dist( a, Z ) (the distance is atta ined since the in tersection of Z with the b all of radius 2dist( a, Z ) around a is compact). There is a segmen t connecting a and z —that is, a metric segment (see the d efinition follo win g the Main Theorem); for R d this simply means a line segment. The segment is a connected set conta ining b oth a ∈ D and z / ∈ D , so it meets ∂ D , and th us also C , at some p oin t, sa y c . Hence, dist( a, z ) = d ist( a, c ) + dist( c, z ) > dist( a, c ) ≥ dist( a, C ). W e also ha ve dist( a, C ) = dist( a, D ) for all a / ∈ D . (10) F or let d ∈ D b e arbitrary . Again, there is a segmen t connecting a and d , and it meets ∂ D , and th us also C , at some p oint, sa y c . He nce, dist( a, d ) = dist( a, c ) + dist( c, d ) ≥ dist( a, c ) ≥ dist( a, C ). Since C ⊆ D , this pr o ves (10). The first part of (a) comes as follo ws: P oints a ∈ D b elong b oth to dom( D, Z ) and, by (9), to d om( C, Z ); other p oin ts a / ∈ D b elong to dom( D , Z ) and d om( C, Z ) at the same time b y (10). The second p art is similar: P oint s a ∈ D b elong neither to dom( Z, D ) nor to dom( Z, C ) b y (9); other p oin ts a / ∈ D b elong to dom( Z, D ) and dom( Z, C ) at the same time by (10). No w w e pro ceed with k -gradations. Let ( R i , S i ) i b e a k -gradation for P and Q as in Prop osition 2. W e ob s erv e that R i ∪ S i is the wh ole sp ace and that P = R 0 ⊆ R 1 ⊆ · · · ⊆ R k − 1 , S 1 ⊇ S 2 ⊇ · · · ⊇ S k = Q, (11) b ecause X ⊆ dom( X, Y ). Lemma 4. L et P , Q b e nonempty, disjoint, c lose d se ts in an arbitr ary metric sp ac e. (i) If ( C i ) i is a k -se ctor of C 0 = P and C k = Q , then C i − 1 and C i +1 ar e disjoint for e ach i = 1 , . . . , k − 1 . (ii) If ( R i , S i ) i is a k - gr adation b etwe en R 0 = P and S k = Q , then R i and S j ar e disjoint for e ach i and j with 0 ≤ i < j ≤ k . Pr o of. Supp ose that there is a p oint a ∈ C i − 1 ∩ C i +1 . Since d ist( a, C i − 1 ) = 0 = dist( a, C i +1 ), w e ha ve a ∈ bisect( C i − 1 , C i +1 ) = C i . Since P and Q are disj oin t, either a / ∈ P or a / ∈ Q . By symmetry , we ma y assume a / ∈ P . Let i − b e the smallest such that a ∈ C j for all j = i − , . . . , i . Then a ∈ C i − +1 \ C i − − 1 , contradict ing a ∈ C i − = bisect( C i − − 1 , C i − +1 ). F or (ii), s u pp ose that there is a p oin t a ∈ R i ∩ S j for some i < j . Since P and Q are d isjoin t, either a / ∈ P or a / ∈ Q . By symmetry , w e may assume a / ∈ P . Retake i to b e the smallest suc h that a ∈ R i . Then a / ∈ R i − 1 and a ∈ S j ⊆ S i +1 , contradict ing a ∈ R i = dom( R i − 1 , S i +1 ). Pr o of of Pr op osition 2. F or one direction, let ( R i , S i ) i b e a k -gradation an d let C i = R i ∩ S i for eac h i = 1, . . . , k − 1. Then C i = dom( R i − 1 , S i +1 ) ∩ dom( S i +1 , R i − 1 ) = b isect( R i − 1 , S i +1 ) is nonemp t y . Moreo ve r, this equals bisect( C i − 1 , C i +1 ) b y Lemm a 3(b), b ecause R i − 1 and S i +1 are disj oin t according to Lemma 4(ii). F or the other direction, we su pp ose that ( C i ) i is a k -sector. Let R i = dom( C i − 1 , C i +1 ) and S i = d om( C i +1 , C i − 1 ) for eac h i = 1, . . . , k − 1. Then C i = R i ∩ S i b y the definition of a k - sector. By Lemma 4(i), we hav e R i ∩ C i +1 = ∅ , and similarly S i +1 ∩ C i = ∅ . Therefore, R i ∩ S i +1 is d isjoin t fr om C i ∪ C i +1 ⊇ ∂ R i ∪ ∂ S i +1 ⊇ ∂ ( R i ∩ S i +1 ). This means that R i ∩ S i +1 has an empt y b oundary , and th us is itself empty , b ecause the whole space is geod esic and hence connected. By this and the fact that R i ∪ S i co vers the whole space, w e ha v e P ⊆ R 1 ⊆ · · · ⊆ R k − 1 and 6 X Y bisect( X, Y ) Figure 4 : A bise c tor may b e fat in the plane with the ℓ 1 metric. Every p oint in the sha ded region is a t the same distance from X and Y . The equation in Lemma 6 do es not hold. S 1 ⊇ S 2 ⊇ · · · ⊇ S k − 1 ⊇ Q . Beca use R i and S i +1 are disjoint , so are R i − 1 and S i +1 . This allo ws us to app ly Lemma 3(a), whic h yields dom( R i − 1 , S i +1 ) = dom( C i − 1 , C i +1 ) = R i and similarly dom( S i +1 , R i − 1 ) = S i . The follo wing example shows that th e assu m ption of the space b eing geo desic cannot b e dropp ed. Consider the distance on R defined by dist( x, y ) = f ( | x − y | ), where f is given by f ( r ) =      r if r ≤ 1 , 1 if 1 ≤ r ≤ 2 , r / 2 if r ≥ 2 . (12) Th us , d is al most lik e the usual metric, except that it “thinks of an y distance b et w een 1 and 2 as the same.” T hen there is no trisector b etw een P = ( −∞ , 0] and Q = [1 , + ∞ ) (whereas th er e is a gradation b y Prop osition 1). F or supp ose that ( C 1 , C 2 ) is a trisector. By Lemma 4(i) , the set C 2 cannot ov erlap P or Q , so it is a nonempt y s ubset of (0 , 1). Hence, the p oin t 2 is equidistan t from C 2 and P , and thus b elongs to C 1 . This con tradicts Lemma 4(i). 4 Dra wing k -sectors Here w e provide a more constructiv e p ro of of the existence of k -gradations, bu t only under stronger assumptions than in Prop osition 1. Later w e discuss h ow this appr oac h can b e used for ap p ro ximate computation of bisectors. W e w r ite X for the closure of a set X . Prop osition 5. Supp ose that P and Q ar e disjoint nonempty close d sets in R d with the Euclide an norm (or, mor e gener al ly, with an arbitr ary strictly c onvex norm). L et the lat- tic e L and the function F : L → L b e as in the the pr o of of Pr op osition 1 (Se ction 2 ). L et ( R 0 i , S 0 i ) i b e an arbitr ary element of L with ( R 0 i , S 0 i ) i ≤ F (( R 0 i , S 0 i ) i ) . Define ( R n +1 i , S n +1 i ) i := F (( R n i , S n i ) i ) for e ach n ∈ N (thus, ( R 0 i , S 0 i ) i ≤ ( R 1 i , S 1 i ) i ≤ ( R 2 i , S 2 i ) i ≤ · · · ), and let ( R ∞ i , S ∞ i ) i = W { ( R n i , S n i ) i : n ∈ N } . Then ( R ∞ i , S ∞ i ) i is a k - gr adation. W e b egin proving this prop osition. W e write R n 0 = P and S n k = Q for eac h n ∈ N ∪ {∞} . Lemma 6. F or any disjoint nonempty close d sets X , Y in R d with the Euclide an metric (or with a strictly c onvex norm ), dom( Y , X ) = R d \ dom( X, Y ) . W e n ote that the assumption on the considered metric in this lemma is necessary: As Figure 4 illustrates, the claim is n ot v alid with the ℓ 1 norm. 7 z y Y X z ′ Figure 5: Since X do es not intersect the in terio r of the ball a r ound z , it do e s not touc h the ball around z ′ . Pr o of of L emma 6. W e h av e dom( Y , X ) ⊇ R d \ dom( X, Y ) b ecause dom( Y , X ) is closed and dom( Y , X ) ∪ dom ( X, Y ) = R d . F or the other inclusion, let z ∈ d om( Y , X ) an d let y b e a closest p oin t in Y to z . Sin ce X d o es not intersect the op en ball with cen tr e z and radiu s dist( y , z ), an y p oin t z ′ 6 = z on the segment z y is strictly closer to y than to X (Fig ur e 5), and th us is not in dom( X , Y ). Sin ce z ′ can b e arb itrarily close to z , we ha ve z ∈ R d \ dom( X, Y ). Lemma 7. If ( R ∞ i , S ∞ i ) i is as in Pr op osition 5, then R ∞ i ∩ S ∞ j = ∅ whenever 0 ≤ i < j ≤ k . Pr o of. F or contradict ion, s u pp ose that there is some a ∈ R ∞ i ∩ S ∞ j . If i > 0, then for eac h n ∈ N w e h a ve a ∈ S ∞ j ⊆ S n j ⊆ S n i +1 , so dom( R n i − 1 , { a } ) ⊇ dom( R n i − 1 , S n i +1 ) = R n +1 i . This implies dist( a, R n i − 1 ) ≤ 2 · d ist( a, R n +1 i ). Since a ∈ R ∞ i , the righ t-hand side tend s to 0 as n → ∞ , and hence, so do es dist( a, R n i − 1 ). Thus, a ∈ R ∞ i − 1 . Rep eating th e same argument for i − 1, i − 2, . . . , we arriv e at a ∈ R ∞ 0 = P . Similarly , if j < k , th en a ∈ S ∞ j ⊆ S n +1 j = dom( S n j +1 , R n j − 1 ) f or all n ∈ N . Th u s, dist( a, S n j +1 ) ≤ dist( a, R n j − 1 ) ≤ dist( a, R n i ) → 0 as n → ∞ b ecause a ∈ R ∞ i . So a ∈ S ∞ j +1 . Rep eating th e argument for j + 1, j + 2, . . . , we obtain a ∈ S ∞ k = Q . Th us w e ha ve a ∈ P ∩ Q , con tradicting the assum p tion that P and Q are disjoint. Pr o of of Pr op osition 5. Our go al is to show th at F (( R ∞ i , S ∞ i ) i ) = ( R ∞ i , S ∞ i ) i . Since F is monotone, F (( R ∞ i , S ∞ i ) i ) ≥ F (( R n i , S n i ) i ) ≥ ( R n i , S n i ) i for eac h n , and h ence F (( R ∞ i , S ∞ i ) i ) ≥ ( R ∞ i , S ∞ i ) i . It remains to sho w that F (( R ∞ i , S ∞ i ) i ) ≤ ( R ∞ i , S ∞ i ) i , w hic h means, b y the d efi- nition of F , that dom( S ∞ i +1 , R ∞ i − 1 ) ⊇ S ∞ i , (13) and dom( R ∞ i − 1 , S ∞ i +1 ) ⊆ R ∞ i . (14) The inclus ion (13) follo ws just by conti nuit y of the d istance fu nction: W e ha ve S ∞ i = T n ∈ N S n +1 i = T n ∈ N dom  S n i +1 , R n i − 1  . So for x ∈ S ∞ i w e h av e d ist( x, S n i +1 ) ≤ d ist( x, R n i − 1 ) for ev ery n , and dist( x, S ∞ i +1 ) = lim n →∞ dist( x, S n i +1 ) ≤ lim n →∞ dist( x, R n i − 1 ) = dist( x, R ∞ i − 1 ). Hence x ∈ dom( S ∞ i +1 , R ∞ i − 1 ) and (13) is prov ed. F or pro ving (14), we need the p revious lemmas. By (13 ), we hav e R d \ dom( S ∞ i +1 , R ∞ i − 1 ) ⊆ R d \ S ∞ i ⊆ R ∞ i , (15) 8 where the latter inclusion is b ecause R n i ∪ S n i = R d for ev ery n (this wa s part of the d efi nition of L ). W e obtain (14) b y taking the closure of (15), using Lemma 6 for the left-hand s id e; for ap p lying this lemma, w e need R ∞ i − 1 ∩ S ∞ i +1 = ∅ , w hic h holds by Lemma 7. If the initial elemen t ( R 0 i , S 0 i ) i in Prop osition 5 is less than or equal to all k -gradations (with r esp ect to the ord ering ≤ ), then so is ( R n i , S n i ) i for all n (inductiv ely by the monotonicit y of F ), and th erefore, the resulting ( R ∞ i , S ∞ i ) i is the le ast k -gradation. Th is is the case w hen, for examp le, ( R 0 i , S 0 i ) i is the least element ( P , R d ) i of L . The trisector in Figure 3 corresp ond s to the least 3-gradatio n, but this 3-gradation is n ot obtained by iteration from the least element of L . This witnesses th at Prop osition 5 may indeed fail for norms that are not strictly conv ex. Computational issues Pr op osition 5 giv es a method to draw a k -sector in Euclidean spaces: By applying F iterativ ely , w e get an a scendin g chain ( R 0 i , S 0 i ) i ≤ ( R 1 i , S 1 i ) i ≤ · · · whose supremum ( R ∞ i , S ∞ i ) i giv es a k -gradation ( R ∞ i , S ∞ i ) i . If we stop the iteration after sufficien tly many steps, w e obtain an appro ximation of ( R ∞ i , S ∞ i ) i . Ho wev er, implementing the algorithm is not entirely tr ivial, b ecause ev en if the sites are simple, applying F rep eatedly give s r ise to regions that are hard to describ e. F or example, consider the case where P and Q are p oin ts in the plane, and we b egin w ith ( R 0 i , S 0 i ) i = ( P , R 2 ) i . Then ∂ R 1 k − 1 is the line bisecting P and Q , and ∂ R 2 k − 2 is the parab ola bisecting P and this line. The next iteration yields the curve ∂ R 3 k − 3 (or ∂ R 3 k − 1 ) wh ic h bisects b et we en a parab ola and a p oint. Th us , unlik e typical basic op erations allo wed in computational geometry , taking the b i- sector giv es rise to increasingly complicate d cur v es. If w e ha v e an analytic description of the b oundary curves of the regions R n i and S n i , eac h of the curves definin g R n +1 i and S n +1 i is describ ed b y a system of d ifferen tial equations asso ciated with the b isecting condition. But solving such equations exactly in eac h iterativ e step is computationally exp ensiv e. Therefore, w e need to fi nd a practical metho d for app ro x im ating th e b isectors (assum ing that we only compute the regions in a b ounded area). One metho d is to appro ximate eac h regio n by a p olygonal region. W e start with some p olygonal appr o ximations ˜ P , ˜ Q of P , Q , and let ( ˜ R 0 i , ˜ S 0 i ) i := ( ˜ P , R d ) i . Th en for eac h n , w e compute an ap p ro ximation ( ˜ R n +1 i , ˜ S n +1 i ) i to F (( ˜ R n i , ˜ S n i ) i ), where th e bisector of t wo p olygonal regions, which is a p iecewise quadratic curve, is appr o ximated b y a suitable p olygonal region. T o ens ure that ( ˜ R n i , ˜ S n i ) i con v erges to an u nderestimate (with resp ect to the ord ering ≤ ) of the least k -gradation ( R ∞ i , S ∞ i ) i , w e should h a ve ( ˜ R n i , ˜ S n i ) i ≤ ( ˜ R n +1 i , ˜ S n +1 i ) i ≤ F (( ˜ R n i , ˜ S n i ) i ). Th is can b e ac h iev ed by computing an inn er app r o xim ation of R n +1 i and an outer app ro ximation of S n +1 i . Another metho d is to consider the p roblem in the pixel geometry , wh ere eac h of the appro ximate r egions ˜ R n i , ˜ S n i is a set of pixels. In computing ( ˜ R n +1 i , ˜ S n +1 i ) i , we aga in mak e sure that ( ˜ R n i , ˜ S n i ) i ≤ ( ˜ R n +1 i , ˜ S n +1 i ) i ≤ F (( ˜ R n i , ˜ S n i ) i ). Th en ( ˜ R n i , ˜ S n i ) i stabilizes even tually , pro viding a lo w er estimate of the least k -gradation. T h e analysis of time complexit y (as a function of pr ecision) of these metho ds is left as a future research pr oblem. Uniqueness The curve s in Figure 1 we re drawn usin g the pixel geomet ry mo del explained ab o v e. As w e menti oned there, they are guaranteed to lie on P ’s side of an y true k -sector curv es. By exchanging P and Q , we obtain also an appro ximate k -sector that lies on Q ’s side of any true k -sector. W e tried compu ting th ese lo wer and upp er estimates f or sev eral 9 differen t P , Q and k in Euclidean p lane, but w e did n ot find them differ by a significan t amoun t. Because of this, w e su sp ect that the k -sector is alw ays unique: Conjecture. The k -se ctor of any two disjoint nonempty close d sets in E u clide an sp ac e is unique. Ac kno wledgemen ts W e gratefully ac knowle dge v aluable discussions with many friends includ ing T etsuo Asano and G ¨ unt er Rote; ind eed, we o we T etsuo for precious information of his r ecen t work on conv ex distance cases. W e also thank Y u Muramatsu for his programming w ork in dra win g figures. D. R. would lik e to express his th anks to Simeon Reic h for his helpfu l discussion. Finally , w e remark th at the wa rm commen ts f rom the audience of the preliminary announ cement at EuroCG 2009 encouraged us to w ork further on th e su b ject. A. K. is su pp orted by the Nak a jima F oundation and the Natur al Sciences and E n gineering Researc h Council of Canada. The part of this researc h by T. T. w as partially supp orted by the JS PS Grant- in-Aid for S cien tifi c Researc h (B) 18300001. References [1] T . Asano and D. Kirkpatrick. Distance trisector cur v es in regular con vex d istance met- rics. 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