Hyperplane Arrangements with Large Average Diameter
aa r X i v : . [ m a t h . M G ] O c t Hyperplane Arrangementswith Large Average Diameter
Antoine Deza and Feng Xie
September 23, 2007McMaster UniversityHamilton, Ontario, Canada deza, xief @mcmaster.ca
Abstract:
The largest possible average diameter of a bounded cell of a simplehyperplane arrangement is conjectured to be not greater than the dimension. Weprove that this conjecture holds in dimension 2, and is asymptotically tight in fixeddimension. We give the exact value of the largest possible average diameter for allsimple arrangements in dimension 2, for arrangements having at most the dimensionplus 2 hyperplanes, and for arrangements having 6 hyperplanes in dimension 3. Indimension 3, we give lower and upper bounds which are both asymptotically equalto the dimension.
Keywords: hyperplane arrangements, bounded cell, average diameter
Let A be a simple arrangement formed by n hyperplanes in dimension d . We recall that anarrangement is called simple if n ≥ d + 1 and any d hyperplanes intersect at a unique distinctpoint. The number of bounded cells (closures of the bounded connected components of thecomplement) of A is I = (cid:0) n − d (cid:1) . Let δ ( A ) denote the average diameter of a bounded cell P i of A ; that is, δ ( A ) = P i = Ii =1 δ ( P i ) I where δ ( P i ) denotes the diameter of P i , i.e., the smallest number such that any two verticesof P i can be connected by a path with at most δ ( P i ) edges. Let ∆ A ( d, n ) denote the largestpossible average diameter of a bounded cell of a simple arrangement defined by n inequalitiesin dimension d . Deza, Terlaky and Zinchenko conjectured that ∆ A ( d, n ) ≤ d . Conjecture 1 [5]
The average diameter of a bounded cell of a simple arrangement defined by m inequalities in dimension n is not greater than n . It was showed in [5] that if the conjecture of Hirsch holds for polytopes in dimension d , then∆ A ( d, n ) would satisfy ∆ A ( d, n ) ≤ d + dn − . In dimension 2 and 3, we have ∆ A (2 , n ) ≤ n − and ∆ A (3 , n ) ≤ n − . We recall that a polytope is a bounded polyhedron and that theconjecture of Hirsch, formulated in 1957 and reported in [1], states that the diameter of aolyhedron defined by n inequalities in dimension d is not greater than n − d . The conjecturedoes not hold for unbounded polyhedra.Conjecture 1 can be regarded a discrete analogue of a result of Dedieu, Malajovich andShub [4] on the average total curvature of the central path associated to a bounded cell of a simplearrangement. We first recall the definitions of the central path and of the total curvature. For apolytope P = { x : Ax ≥ b } with A ∈ ℜ n × d , the central path corresponding to min { c T x : x ∈ P } is a set of minimizers of min { c T x + µf ( x ) : x ∈ P } for µ ∈ (0 , ∞ ) where f ( x ) = − P ni =1 ln( A i x − b i ) – the standard logarithmic barrier function [12]. Intuitively, the total curvature [14] is ameasure of how far off a certain curve is from being a straight line. Let ψ : [ α, β ] → ℜ d be a C (( α − ε, β + ε )) map for some ε > α, β ]. Denote its arc lengthby l ( t ) = R tα k ˙ ψ ( τ ) k dτ , its parametrization by the arc length by ψ arc = ψ ◦ l − : [0 , l ( β )] → ℜ d ,and its curvature at the point t by κ ( t ) = ¨ ψ arc ( t ). The total curvature is defined as R l ( β )0 k κ ( t ) k dt .The requirement ˙ ψ = 0 insures that any given segment of the curve is traversed only once andallows to define a curvature at any point on the curve. Let λ c ( A ) denote the average associatedtotal curvature of a bounded cell P i of a simple arrangement A ; that is, λ c ( A ) = i = I X i =1 λ c ( P i ) I where λ c ( P ) denotes the total curvature of the central path corresponding to the linear op-timization problem min { c T x : x ∈ P } . Dedieu, Malajovich and Shub [4] demonstrated that λ c ( A ) ≤ πd for any fixed c . Keeping the linear optimization approach but replacing centralpath following interior point methods by simplex methods, Haimovich’s probabilistic analysis ofthe shadow-vertex simplex algorithm, see [2, Section 0.7], showed that the expected number ofpivots is bounded by d . Note that while Dedieu, Malajovich and Shub consider only the boundedcells (the central path may not be defined over some unbounded ones), Haimovich considers theaverage over bounded and unbounded cells. While the result of Haimovich and Conjecture 1 aresimilar in nature, they differ in some aspects: Conjecture 1 considers the average over boundedcells, and the number of pivots could be smaller than the diameter for some cells.In Section 4 we consider a simple hyperplane arrangement A ∗ d,n combinatorially equivalentto the cyclic hyperplane arrangement which is dual to the cyclic polytope, see [8] for somecombinatorial properties of the (projective) cyclic hyperplane arrangement. We show that thebounded cells of A ∗ d,n are mainly combinatorial cubes and, therefore, that the dimension d is anasymptotic lower bound for ∆ A ( d, n ) for fixed d . In Section 2, we consider the arrangement A o ,n resulting from the addition of one hyperplane to A ∗ ,n − such that all the vertices are on one sideof the added hyperplane. We show that the arrangement A o ,n maximizes the average diameterand, thus, Conjecture 1 holds in dimension 2. In Section 3, considering a 3-dimensional analogue,we give lower and upper bounds asymptotically equal to 3 for ∆ A (3 , n ). The combinatorics ofthe addition of a (pseudo) hyperplane to the cyclic hyperplane arrangement is studied in detailsin [16]. For example, the arrangements A ∗ , and A o , correspond to the top and bottom elementsof the higher Bruhat order B (5 ,
2) given in Figure 3 of [16]. For polytopes and arrangements,we refer to the books of Edelsbrunner [6], Gr¨unbaum [10] and Ziegler [17].2
Line Arrangements with Maximal Average Diameter
For n ≥
4, we consider the simple line arrangement A o ,n made of the 2 lines h and h forming,respectively, the x and x axis, and the ( n −
2) lines defined by their intersections with h and h . We have h k ∩ h = { k − ε, } and h k ∩ h = { , − ( k − ε } for k = 3 , , . . . , n − h n ∩ h = { , } and h n ∩ h = { , ε } where ε is a constant satisfying 0 < ε < / ( n − A o , . h h Figure 1: An arrangement combinatorially equivalent to A o , Proposition 2
For n ≥ , the bounded cells of the arrangement A o ,n consist of ( n − triangles, ( n − n − n -gon. We have δ ( A o ,n ) = 2 − ⌈ n ⌉ ( n − n − for n ≥ . Proof:
The first ( n −
1) lines of A o ,n clearly form a simple line arrangement A ∗ ,n − whichbounded cells are ( n −
3) triangles and (cid:0) n − (cid:1) h n adds 1 n -gon, 1 triangle3nd ( n −
4) 4-gons. Since the diameter of a k -gon is ⌊ k ⌋ , we have δ ( A o ,n ) = 2 − ( n − − ( ⌊ n ⌋− n − n − =2 − ⌈ n ⌉ ( n − n − . (cid:3) Exploiting the fact that a line arrangement contains at least n − n − d simplices for a simple hyperplane arrangement [13]) and a bound on the number of facets on theboundary of the union of the bounded cells, we can show that A o ,n attains the largest possibleaverage diameter of a simple line arrangement. Proposition 3
For n ≥ , the largest possible average diameter of a bounded cell of a simpleline arrangement satisfies ∆ A (2 , n ) = 2 − ⌈ n ⌉ ( n − n − . Proof:
Let f ( A ) denote the number of bounded edges of a simple arrangement A of n lines,and let f ( P i ) denote the number of edges of a bounded cell P i of A . Let call an edge of A external if it belongs to exactly one bounded cell, and let f ( A ) denote the number of externaledges of A . Let p odd ( A ) be the number of bounded cells having an odd number of edges. Wehave: I × δ ( A ) = I X i =1 δ ( P i ) = I X i =1 (cid:22) f ( P i )2 (cid:23) = I X i =1 f ( P i )2 − p odd ( A )2 = 2 f ( A ) − f ( A ) − p odd ( A )2 . Since f ( A ) = n ( n − δ ( A ) is equivalent to minimize f ( A ) + p odd ( A ). We clearlyhave f ( A o ,n ) = 2( n − f ( A )is at least 2( n − p odd ( A o ,n ) = n − n , and this is the bestpossible since at least n − p odd ( A )is odd, P Ii =1 f ( P i ) is odd. If f ( A o ,n ) = 2( n − P Ii =1 f ( P i ) = 2 f ( A ) − f ( A ) is even. Thus,for odd n , f ( A ) + p odd ( A ) is at least 2( n −
1) + ( n −
2) + 1 which is achieved by A o ,n . Thus A o ,n minimizes f ( A ) + p odd ( A ); that is, maximizes δ ( A ). (cid:3) For n ≥
5, we consider the simple plane arrangement A o ,n made of the the 3 planes h , h and h corresponding, respectively, to x = 0, x = 0 and x = 0, and ( n −
3) planes defined bytheir intersections with the x , x and x axis. We have h k ∩ h ∩ h = { k − ε, , } , h k ∩ h ∩ h = { , k − ε, } and h k ∩ h ∩ h = { , , − ( k − ε } for k = 4 , , . . . , n − h n ∩ h ∩ h = { , , } , h n ∩ h ∩ h = { , , } and h n ∩ h ∩ h = { , , ε } where ε is a constant satisfying 0 < ε < / ( n − A o , where, for clarity, only the bounded cells belonging to thepositive orthant are drawn. Proposition 4
For n ≥ , the bounded cells of the arrangement A o ,n consist of ( n − tetra-hedra, ( n − n − − cells combinatorially equivalent to a prism with a triangular base, (cid:0) n − (cid:1) cells combinatorially equivalent to a cube, and cell combinatorially equivalent to ashell S n with n facets and n − vertices. See Figure 3 for an illustration of S . We have δ ( A o ,n ) = 3 − n − + ⌊ n ⌋− n − n − n − for n ≥ . h h Figure 2: An arrangement combinatorially equivalent to A o , Proof:
For 4 ≤ k ≤ n −
1, let A ∗ ,k denote the arrangement formed by the first k planesof A o ,n . See Figure 4 for an arrangement combinatorially equivalent to A ∗ , . We first showby induction that the bounded cells of the arrangement A ∗ ,n − consist of ( n −
4) tetrahedra,( n − n −
5) combinatorial triangular prisms and (cid:0) n − (cid:1) combinatorial cubes. We use thefollowing notation to describe the bounded cells of A ∗ ,k − : T △ for a tetrahedron with a faceton h ; P △ , respectively P ⋄ , for a combinatorial triangular prism with a triangular, respectivelysquare, facet on h ; C ⋄ for a combinatorial cube with a square facet on h ; and C , respectively T and P , for a combinatorial cube, respectively tetrahedron and triangular prism, not touching h . When the plane h k is added, the cells T △ , P △ , P ⋄ , and C ⋄ are sliced, respectively, into T and P △ , P and P △ , P and C ⋄ , and C and C ⋄ . In addition, one T △ cell and ( k − P ⋄ cells are created by bounding ( k −
3) unbounded cells of A ∗ ,k − . Let c ( k ) denotes the numberof C cells of A ∗ ,k , similarly for C ⋄ , T , T △ , P , P △ and P ⋄ . For A ∗ , we have t △ (4) = 1 and t (4) = p (4) = p △ (4) = p ⋄ (4) = c (4) = c (4) = 0. The addition of h k removes and adds one T △ ,thus, t △ ( k ) = 1. Similarly, all P ⋄ are removed and ( k −
4) are added, thus, p ⋄ ( k ) = ( k − S Since t ( k ) = t ( k −
1) + t △ ( k −
1) and p △ ( k ) = p △ ( k −
1) + t △ ( k − t ( k ) = p △ ( k ) =( k − p ( k ) = p ( k −
1) + p △ ( k −
1) + p ⋄ ( k − p ( k ) = ( k − k − c ⋄ ( k ) = c ⋄ ( k −
1) + p ⋄ ( k − c ⋄ ( k ) = (cid:0) k − (cid:1) . Since c ( k ) = c ( k −
1) + c ⋄ ( k − c ( k ) = (cid:0) k − (cid:1) . Therefore the bounded cells of A ∗ ,n − consist of t ( n −
1) + t △ ( n −
1) = ( n − p ( n −
1) + p △ ( n −
1) + p ⋄ ( n −
1) = ( n − n −
5) combinatorial triangular prisms,and c ( n −
1) + c ⋄ ( n −
1) = (cid:0) n − (cid:1) combinatorial cubes. The addition of h n to A ∗ ,n − creates1 shell S n with 2 triangular facets belonging to h and h and 1 square facet belonging to h . Besides S n , all the bounded cells created by the addition of h n are below h . One P ⋄ and n − h and h . The other bounded cells areon the negative side of h : n − P ⋄ and 1 T △ between h n and h n − , and n − k − C ⋄ and1 P △ between h n − k and h n − k − for k = 1 , . . . , n −
5. In total, we have 1 tetrahedron, (cid:0) n − (cid:1) combinatorial cubes and (2 n −
9) combinatorial triangular prisms below h . Since the diameterof a tetrahedron, triangular prism, cube and n -shell is, respectively, 1 , , ⌊ n ⌋ , we have δ ( A o ,n ) = 3 − n − n − n − − − ( ⌊ n ⌋− n − n − n − = 3 − n − + ⌊ n ⌋− n − n − n − . (cid:3) Remark 5
There is only one combinatorial type of simple arrangement of planes, and we have ∆ A (3 ,
5) = δ ( A o , ) = . Among the simple combinatorial types of arrangements formed by planes [7], the maximum average diameter is while δ ( A o , ) = 1 . . See Figure 5 for an illustra-tion of the combinatorial type of one of the two simple arrangements with planes maximizingthe average diameter. The far away vertex on the right and bounded edges incident to it arecut off (same for the far away vertex on the left) so the bounded cells of the arrangement ( tetrahedra, simplex prisms, and Proposition 6
For n ≥ , the largest possible average diameter of a bounded cell of a simplearrangement of n planes satisfies − n − + ⌊ n ⌋− n − n − n − ≤ ∆ A (3 , n ) ≤ n − n +21)3( n − n − n − . Proof:
Let f ( A ) denote the number of bounded facets of a simple arrangement A of n planes,and let f ( P i ) denote the number of facets of a bounded cell P i of A . Let call a facet of A external if it belongs to exactly one bounded cell, and let f ( A ) denote the number of externalfacets of A . We have: I × δ ( A ) = I X i =1 δ ( P i ) ≤ I X i =1 (cid:18)(cid:22) f ( P i )3 (cid:23) − (cid:19) ≤ I X i =1 f ( P i )3 − n − − I = 4 f ( A ) − f ( A ) − n + 3 − I n −
3) bounded cells of A are simplices [13].Since f ( A ) = n (cid:0) n − (cid:1) and f ( A ) is at least n ( n − + 2, see [3], we have δ ( A ) ≤ n − n +21) / n − n − n − (cid:3) h h Figure 4: An arrangement combinatorially equivalent to A ∗ , After recalling in Section 4.1 the unique combinatorial structure of a simple arrangement formedby d + 2 hyperplanes in dimension d , we show in Section 4.2 that the cyclic hyperplane arrange-ment A ∗ d,n contains (cid:0) n − dd (cid:1) cubical cells for n ≥ d . It implies that the average diameter δ ( A ∗ d,n )is arbitrarily close to d for n large enough. Thus, the dimension d is an asymptotic lower boundfor ∆ A ( d, n ) for fixed d . d + 2 hyperplanes Let A d,d +2 be a simple arrangement formed by d + 2 hyperplanes in dimension d . Besidessimplices, the bounded cells of A d,d +2 are simple polytopes with d + 2 facets corresponding tothe product of a k -simplex with a ( d − k )-simplex for k = 1 , . . . , ⌊ d ⌋ , see for example [10]. Werecall one way to show that the combinatorial type of the arrangement of d + 2 hyperplanes indimension d is unique. The affine Gale dual, see [16, Chapter 6], of the d +3 vectors in dimension d + 1 corresponding to the linear arrangement associated to A d,d +2 (and the hyperplane atinfinity) forms a configuration of d + 3 distinct signed points on a line; i.e., is unique up torelabeling and reorientation. We also recall the combinatorial structure of A d,d +2 as some of the7igure 5: An arrangement formed by 6 planes maximizing the average diameternotions presented are used in Section 4.2. Since there is only one combinatorial type of simplearrangement with d + 2 hyperplanes, the arrangement A d,d +2 can be obtained from the simplex A d,d +1 by cutting off one its vertices v with the hyperplane h d +2 . As a result, a prism P witha simplex base is created. Let us call top base the base of P which belongs to h d +2 and assume,without loss of generality, that the hyperplane containing the bottom base of P is h d +1 . Besidesthe simplex defined by v and the vertices of the top base of P , the remaining d bounded cells of A d,d +2 are between h d +2 and h d +1 . See Figure 6 for an illustration the combinatorial structureof A , . As the projection of A d,d +2 on h d +1 is combinatorially equivalent to A d − ,d +1 , the d bounded cells between h d +2 and h d +1 can be obtained from the d bounded cells of A d − ,d +1 bythe shell-lifting of A d − ,d +1 over the ridge h d +1 ∩ h d +2 ; that is, besides the vertices belongingto h d +1 ∩ h d +2 , all the vertices in h d +1 (forming A d − ,d +1 ) are lifted. See Figure 7 where theskeletons of the d + 1 bounded cells of A d,d +2 are given for d = 2 , , . . . ,
6, and the shell-lifting ofthe bounded cells is indicated by an arrow. The vertices not belonging to h d +1 are representedin black in Figure 7, e.g., the simplex cell containing v is the one made of black vertices. Thebounded cells of A d,d +2 are 2 simplices and a pair of product of a k -simplex with a ( d − k )-simplex for k = 1 , . . . , ⌊ d ⌋ for odd d . For even d the product of the d -simplex with itself ispresent only once. Since all the bounded cells, besides the 2 simplices, have diameter 2, we have δ ( A d,d +2 ) = d − d +1 . Proposition 7
We have ∆ A ( d, d + 2) = δ ( A d,d +2 ) = dd +1 . We consider the simple hyperplane arrangement A ∗ d,n combinatorially equivalent to the cyclichyperplane and formed by the following n hyperplanes h dk for k = 1 , , . . . , n . The hyperplanes h dk = { x : x d +1 − k = 0 } for k = 1 , , . . . , d form the positive orthant, and the hyperplanes h dk for k = d + 1 , . . . , n are defined by their intersections with the axes ¯ x i of the positiveorthant. We have h dk ∩ ¯ x i = { , . . . , , d − i )( k − d − ε, , . . . , } for i = 1 , . . . , d − A , h dk ∩ ¯ x d = { , . . . , , − ( k − d − ε } where ε is a constant satisfying 0 < ε < / ( n − d − A ∗ d,n can be derived inductively. All the bounded cells of A ∗ d,n are onthe positive side of h d and h d with the bounded cells between h d and h d being obtained by theshell-lifting of a combinatorial equivalent of A ∗ d − ,n − over the ridge h d ∩ h d , and the boundedcells on the other side of h d forming a combinatorial equivalent of A ∗ d,n − . The intersection A ∗ d,n ∩ h dk is combinatorially equivalent to A ∗ d − ,n − for k = 2 , , . . . , d and removing h d from A ∗ d,n yields an arrangement combinatorially equivalent to A ∗ d,n − . See Figure 4 for an arrangementcombinatorially equivalent to A ∗ , . Proposition 8
The arrangement A ∗ d,n contains (cid:0) n − dd (cid:1) cubical cells for n ≥ d . We have δ ( A ∗ d,n ) ≥ d (cid:0) n − dd (cid:1) / (cid:0) n − d (cid:1) for n ≥ d . It implies that for d fixed, ∆ A ( d, n ) is arbitrarily closeto d for n large enough. Proof:
The arrangements A ∗ n, and A ∗ n, contain, respectively, (cid:0) n − (cid:1) and (cid:0) n − (cid:1) cubical cells.The arrangement A ∗ d, d has 1 cubical cell. Since A ∗ d,n is obtained inductively from A ∗ d,n − bylifting A ∗ d − ,n − over the ridge h d ∩ h d , we count separately the cubical cells between h d and h d and the ones on the other side of h d . The ridge h d ∩ h d is an hyperplane of the arrangements A ∗ d,n ∩ h d and A ∗ d,n ∩ h d which are both combinatorially equivalent to A ∗ d − ,n − . Removing h d − from A ∗ d,n ∩ h d yields an arrangement combinatorially equivalent to A ∗ d − ,n − . It implies that (cid:0) ( n − − ( d − d − (cid:1) cubical cells of A ∗ d,n ∩ h d are not incident to the ridge h d ∩ h d . The shell-lifting ofthese (cid:0) n − d − d − (cid:1) cubical cells (of dimension d −
1) creates (cid:0) n − d − d − (cid:1) cubical cells between h d and h d .As removing h d from A ∗ d,n yields an arrangement combinatorial equivalent to A ∗ d,n − , there are (cid:0) n − − dd (cid:1) cubical cells on the other side of h d . Thus, A ∗ d,n contains (cid:0) n − d − d − (cid:1) + (cid:0) n − d − d (cid:1) = (cid:0) n − dd (cid:1) cubical cells. (cid:3) Proposition 8 can be slightly strengthened to the following proposition.9 =2 d=3 d=4 d=5 d=6
Figure 7: The skeletons of the d + 1 bounded cells of A d,d +2 for d = 2 , , . . . , Proposition 9
Besides (cid:0) n − dd (cid:1) cubical cells, the arrangement A ∗ d,n contains ( n − d ) simplicesand ( n − d )( n − d − bounded cells combinatorially equivalent to a prism with a simplex basefor n ≥ d . We have ∆ A ( d, n ) ≥ ( d − ( n − dd ) +( n − d )( n − d − ( n − d ) for n ≥ d . Proof:
Similarly to Proposition 8, we can inductively count ( n − d ) simplices and ( n − d )( n − d −
1) bounded cells of A ∗ d,n combinatorially equivalent to a prism with a simplex base. Wehave ( n − − ( d −
1) simplices in A ∗ d,n ∩ h d and, since removing h d − from A ∗ d,n ∩ h d yieldsan arrangement combinatorially equivalent to A ∗ d − ,n − , only one of these ( n − d ) simplices of A ∗ d,n ∩ h d is incident to the ridge h d ∩ h d . Thus, between h d and h d , we have 1 simplex incident tothe ridge h d ∩ h d and ( n − d −
1) cells combinatorially equivalent to a prism with a simplex base notincident to the ridge h d ∩ h d . In addition, ( n − d −
1) cells combinatorially equivalent to a prismwith a simplex base are incident to the ridge h d ∩ h d and between h d and h d . These ( n − d − A ∗ d,d +1 by h dk for k = d + 2 , d + 3 , . . . , n . Thus,we have 2( n − d −
1) cells combinatorially equivalent to a prism with a simplex base between h d and h d . Since the other side of h d is combinatorially equivalent to A ∗ n − ,d , it contains ( n − − d )simplices and ( n − d − n − d −
2) bounded cells combinatorially equivalent to a prism with asimplex base. Thus, A ∗ d,n has ( n − d − n − d −
2) + 2( n − d −
1) = ( n − d )( n − d −
1) cellscombinatorially equivalent to a prism with a simplex base and ( n − d ) simplices. As a prismwith a simplex base has diameter 2 and the diameter of a bounded cell is at least 1, we have δ ( A ∗ d,n ) ≥ d ( n − dd ) +2( n − d )( n − d − ( n − d ) − ( n − dd ) − ( n − d )( n − d − ( n − d ) for n ≥ d . (cid:3) Acknowledgments
The authors would like to thank Komei Fukuda, Hiroki Nakayama, andChristophe Weibel for use of their codes [9, 11, 15] which helped to investigate small simplearrangements. The authors are grateful to the anonymous referees for many helpful suggestionsand for pointing out relevant papers [2, 16]. Research supported by an NSERC Discovery grant,a MITACS grant and the Canada Research Chair program.
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