The complexity of the envelope of line and plane arrangements

The complexit y of the en v elop e of line and plane arrangemen ts Da vid Bremner, An toine Deza and F eng Xie Septem b er 15, 2007 Abstract A facet of an h yp erplane arrangemen t is called external if it b elongs to exactly one bounded cell. The set of all external facets forms the env elop e of the arrangemen t. The num b er of external facets of a simple arrangemen t defined b y n h yperplanes in dimension d is h yp othesized to be at least d ` n − 2 d − 1 ´ . In this note w e sho w that, for simple arrangements of 4 lines or more, the minim um n umber of external facets is equal to 2( n − 1), and for simple arrangemen ts of 5 planes or more, the minimum num b er of external facets is betw een n ( n − 2)+6 3 and ( n − 4)(2 n − 3) + 5. 1 In tro duction Let A d,n b e a simple arrangemen t formed by n hyperplanes in dimension d . W e recall that an arrangemen t is called simple if n ≥ d + 1 and an y d h yp erplanes intersect at a distinct p oin t. The closures of connected components of the complement of the h yp erplanes forming A d,n are called the cells, or d -faces, of the arrangemen t. F or k = 0 , . . . , d − 1, the k -faces of A d,n are the k -faces of its cells. A facet is a ( d − 1)-face of A d,n , and a facet b elonging to exactly one bounded cell is called an external facet. Equiv alen tly , an external facet is a b ounded facet whic h b elongs to an un b ounded cell. F or k = 0 , . . . , d − 2, an external k -face is a k -face b elonging to an external facet. Let f 0 k ( A d,n ) denote the num b er of external k -faces of A d,n . The set of all external facets forms the env elop e of the arrangement. It was hypothesized in [1] that an y simple arrangement A d,n has at least d  n − 2 d − 1  external facets. In Section 2, we sho w that a simple arrangement of n lines has at least 2( n − 1) external facets for n ≥ 4, and that this b ound is tight. In section 3, we sho w that a simple arrangement of n planes has at least n ( n − 2)+6 3 external facets for n ≥ 5, and exhibit a simple plane arrangemen t with ( n − 4)(2 n − 3) + 5 external facets. F or p olytop es and arrangements, w e refer to the b ooks of Edelsbrunner [3], Gr¨ un baum [6] and Ziegler [7] and the references therein. 2 The complexit y of the env elop e of line arrangemen ts 2.1 A lo wer b ound Prop osition 2.1. F or n ≥ 4 , a simple line arr angement has at le ast 2( n − 1) external fac ets. Pr o of. The external vertices of a line arrangement can b e divided in to three types, namely v 2 , v 3 and v 4 , corresp onding to external v ertices resp ectiv ely inciden t to 2, 3, and 4 b ounded edges. Let us assign to eac h external vertex v a weigh t of 1 and redistribute it to the 2 lines intersecting at v the follo wing w ay: If v is inciden t to exactly 1 unbounded edge, then give w eight 1 to the line containing 1 Da vid Bremner, Antoine Deza and F eng Xie 2 this edge, and w eight 0 to the other line con taining v ; if v is incident to 2 or 0 un b ounded edges, then give w eight 0 . 5 to eac h of the 2 lines intersecting at v . See Figure 1 for an illustration of the w eight distribution. A total of f 0 0 ( A 2 ,n ) weigh ts is distributed and we can also count this quan tity line-wise. The end vertices of a line b eing of type v 2 or v 3 , w e hav e three types of lines, h 2 , 2 , h 2 , 3 and h 3 , 3 , according to the p ossible t yp es of their end-vertices. As a line of type h 3 , 3 con tains 2 v ertices of type v 3 , its w eight is at least 2. Similarly the w eight of a line of type h 2 , 2 w eight is at least 1. Remarking that a line of type h 2 , 3 con tains at least one vertex of type v 4 yields that the w eight of a line of t yp e h 2 , 3 is at least 1 + 0 . 5 + 0 . 5 = 2. F or n ≥ 4 the num b er of lines of type h 2 , 2 is at most 2 as otherwise the env elop e would b e conv ex which is impossible, see for example [4]. Therefore, counting the total distributed weigh t line-wise, w e ha ve f 0 0 ( A 2 ,n ) ≥ 2 n − 2. Since for a line arrangement the n umber of external facets f 0 1 ( A 2 ,n ) is equal to the num b er of external vertices f 0 0 ( A 2 ,n ), we hav e f 0 1 ( A 2 ,n ) ≥ 2( n − 1). v 2 v 3 v 4 1 1 2 1 2 1 2 1 2 Figure 1: The weigh t distribution for the lines of an arrangemen t (the shaded area corresp onds to the b ounded cells). 2.2 A line arrangemen t attaining the lo w er b ound F or n ≥ 4, consider the follo wing simple line arrangement: A o 2 ,n is made of the 2 lines h 1 and h 2 forming, resp ectively , the x 1 and x 2 axis, and ( n − 2) lines defined b y their in tersections with h 1 and h 2 . W e ha ve h k ∩ h 1 = { 1 + ( k − 3) ε, 0 } and h k ∩ h 2 = { 0 , 1 − ( k − 3) ε } for k = 3 , 4 , . . . , n − 1, and h n ∩ h 1 = { 2 , 0 } and h n ∩ h 1 = { 0 , 2 + ε } where ε is a constant satisfying 0 < ε < 1 / ( n − 3). See Figure 2 for an arrangement combinatorially equiv alent to A o 2 , 7 . One can easily chec k that A o 2 , 7 has 2( n − 1) external facets and therefore the low er b ound given in Prop osition 2.1 is tight. Prop osition 2.2. F or n ≥ 4 , the minimum p ossible numb er of external fac ets of a simple line arr angement is 2( n − 1) . 3 The complexit y of the env elop e of plane arrangemen ts 3.1 A lo wer b ound Prop osition 3.1. F or n ≥ 5 , a simple plane arr angement has at le ast n ( n − 2)+6 3 external fac ets. Pr o of. Let h i for i = 1 , 2 , . . . , n b e the planes forming the arrangement A 3 ,n . F or i = 1 , 2 , . . . , n , the external vertices of the line arrangement A 3 ,n ∩ h i are external vertices of the plane arrange- men t A 3 ,n . F or n ≥ 5, the line arrangemen t A 3 ,n ∩ h i has at least 2( n − 2) external facets by Prop osition 2.1, i.e., at least 2( n − 2) external vertices. Since an external v ertex of A 3 ,n b elongs to 3 planes, it is coun ted three times. In other words, the num b er of external v ertices of A 3 ,n satisfies Da vid Bremner, Antoine Deza and F eng Xie 3 h 1 h 2 Figure 2: An arrangement combinatorially equiv alent to A o 2 , 7 f 0 0 ( A 3 ,n ) ≥ 2 n ( n − 2) 3 for n ≥ 5. As the union of all of the b ounded cells is a piecewise linear ball, see [2], the Euler c haracteristic of the b oundary giv es f 0 0 ( A 3 ,n ) − f 0 1 ( A 3 ,n ) + f 0 2 ( A 3 ,n ) = 2. Since an external vertex belong to at least 3 external edges, we hav e 2 f 0 1 ( A 3 ,n ) ≥ 3 f 0 0 ( A 3 ,n ). Thus, we ha ve 2 f 0 2 ( A 3 ,n ) ≥ f 0 0 ( A 3 ,n ) + 4. As f 0 0 ( A 3 ,n ) ≥ 2 n ( n − 2) 3 , it gives f 0 2 ( A 3 ,n ) ≥ n ( n − 2)+6 3 3.2 A plane arrangemen t with few external facets F or n ≥ 5, w e consider following simple plane arrangement: A o 3 ,n is made of the 3 planes h 1 , h 2 and h 3 corresp onding, resp ectiv ely , to x 3 = 0, x 2 = 0 and x 1 = 0, and ( n − 3) planes defined b y their intersections with the x 1 , x 2 and x 3 axis. W e ha ve h k ∩ h 1 ∩ h 2 = { 1 + 2( k − 4) ε, 0 , 0 } , h k ∩ h 1 ∩ h 3 = { 0 , 1 + ( k − 4) ε, 0 } and h k ∩ h 2 ∩ h 3 = { 0 , 0 , 1 − ( k − 4) ε } for k = 4 , 5 , . . . , n − 1, and h n ∩ h 1 ∩ h 2 = { 3 , 0 , 0 } , h n ∩ h 1 ∩ h 3 = { 0 , 2 , 0 } and h n ∩ h 2 ∩ h 3 = { 0 , 0 , 3 + ε } where ε is a constant satisfying 0 < ε < 1 / ( n − 4). See Figure 3 for an illustration of an arrangemen t combinatorially equiv alent to A o 3 , 7 where, for clarity , only the b ounded cells b elonging to the p ositiv e orthan t are dra wn. W e first c heck by induction that the arrangement A ∗ 3 ,n formed by the first n planes of A o n +1 , 3 has 2( n − 2)( n − 3) external facets. The arrangement A ∗ 3 ,n is combinatorially equiv alent to the plane cyclic arrangement which is dual to the cyclic p olytope, see [5] for com binatorial prop erties of the (pro jective) cyclic arrangemen t in general dimension. See Figure 4 for an illustration of A ∗ 3 , 6 . Let H + 3 denote the half-space defined by h 3 and containing the p ositiv e orthant, and H − 3 the Da vid Bremner, Antoine Deza and F eng Xie 4 h 1 h 2 h 3 Figure 3: An arrangement combinatorially equiv alent to A o 3 , 7 other half-space defined by h 3 . The union of the bounded cells of A ∗ 3 ,n in H − 3 is combinatorially equiv alent to the b ounded cells of A ∗ 3 ,n − 1 and therefore has 2( n − 3)( n − 4) facets on its b oundary b y induction hypothesis, including  n − 3 2  b ounded facets contained in h 3 . These  n − 3 2  b ounded facets also b elong to a b ounded cell of A ∗ 3 ,n in H − 3 and therefore are not external facets of A ∗ 3 ,n . Thus, the num b er of external facets of A ∗ 3 ,n b elonging to a bounded cell in H − 3 is 2( n − 3)( n − 4) −  n − 3 2  . The union of the b ounded cells of A ∗ 3 ,n in H + 3 can b e viewed as a simplex cut by n − 4 sliding do wn planes. It has 2  n − 2 2  + 2( n − 3) = n ( n − 3) facets on its b oundary , including the  n − 3 2  b ounded facets con tained in h 3 b elonging to a bounded cell of A ∗ 3 ,n in H − 3 . Thus, the n umber of external facets of A ∗ 3 ,n b elonging to a b ounded cell in H + 3 is n ( n − 3) −  n − 3 2  . Therefore, A ∗ 3 ,n has n ( n − 3) + 2( n − 3)( n − 4) − 2  n − 3 2  = 2( n − 2)( n − 3) external facets. W e now consider how the addition of h n to A ∗ 3 ,n − 1 impacts the n umber of external facets. This impact is similar in nature to Da vid Bremner, Antoine Deza and F eng Xie 5 the addition of h n to the first n − 1 lines of A o 2 ,n . The addition of h n creates  n 2  new b ounded cells: one ab o ve h 1 that we call the n -shel l , and the other ones b eing b elo w h 1 . The n -shell turns n − 4 external facets of A ∗ 3 ,n − 1 ab o v e h 1 in to internal facets of A o 3 ,n , and adds 3 external facets. F or each external facet of A ∗ 3 ,n − 1 b elonging to h 1 whic h is turned into an internal facet of A o 3 ,n , one external facet of A o 3 ,n on h n and not incident to h 1 is added. Belo w h 1 , the addition of h n creates 3( n − 4) + 2 new external facets of A o 3 ,n with an edge on h 1 . Finally , n − 4 new external facets b elonging to h 1 and b ounded b y h n are created from un b ounded facets of A ∗ 3 ,n − 1 . Thus, the total n umber of external facets of A o 3 ,n is 2( n − 3)( n − 4) − ( n − 4) + 3 + (3( n − 4) + 2) + ( n − 4) = ( n − 4)(2 n − 3) + 5. h 1 h 2 h 3 Figure 4: An arrangement combinatorially equiv alent to A ∗ 3 , 6 Remark 3.1. We do not b elieve that A o 3 ,n minimizes the numb er of external fac ets. Among the 43 simple c ombinatorial typ es of arr angements forme d by 6 planes, the minimum numb er of external fac ets is 22 while A o 3 , 6 has 23 external fac ets. Se e Figur e 5 for an il lustr ation of the c ombinatorial typ e of one of the two simple arr angements with 6 planes having 22 external fac ets. The far away vertex on the right and 3 b ounde d e dges incident to it ar e cut off (same for the far away vertex on the left) so the 10 b ounde d c el ls of the arr angement app e ar not to o smal l. Da vid Bremner, Antoine Deza and F eng Xie 6 Figure 5: An arrangement formed by 6 planes and ha ving 22 external facets Ac knowledgmen ts Research supported by NSER C Discov ery grants, by MIT ACS gran ts, by the Canada Research Chair program, and by the Alexander v on Humboldt F oundation. References [1] A. Deza, T. T erlaky and Y. Zinchenk o: Polytopes and arrangements : diameter and curv ature. Op er ations R ese ar ch L etters (to appear). [2] X. Dong. The b ounded complex of a uniform affine oriented matroid is a ball. Journal of Combinatorial The ory Series A (to appear) [3] H. Edelsbrunner: Algorithms in Combinatorial Ge ometry Springer-V erlag (1987). [4] D. Eu, E. Gu´ evremont and G. T. T oussaint: On en velopes of arrangemen ts of lines. Journal of A lgorithms 21 (1996) 111–148. [5] D. F orge and J. L. Ram ´ ırez Alfons ´ ın: On counting the k -face cells of cyclic arrangemen ts. Eur op e an Journal of Combinatorics 22 (2001) 307–312. [6] B. Gr ¨ un baum: Convex Polytop es . V. Kaib el, V. Klee and G. Ziegler (eds.), Graduate T exts in Mathematics 221, Springer-V erlag (2003). [7] G. Ziegler: L e ctur es on Polytop es . Graduate T exts in Mathematics 152, Springer-V erlag (1995). Da vid Bremner, Antoine Deza and F eng Xie 7 Da vid Bremner F a cul ty of Computer Science, University of New Br unswick, New Brunswick, Canad a . Email : bremner @ un b.ca An toine Deza, F eng Xie Dep ar tment of Computing and Softw are, McMaster University, Hamil ton, Ont ario, Canada . Email : deza, xief @ mcmaster.ca