Simultaneous packing and covering in sequence spaces

SIMUL T ANEO US P ACKING AND COVERING IN SEQUENCE SP ACES KONRAD J. SW ANEPOEL Dedicated to the mem o ry of V ic tor Klee A B S T R A C T . W e adapt a construction of Kle e (19 81) to find a packing of unit balls in ℓ p ( 1 ≤ p < ∞ ) which is efficient i n the sense that enlarging the r adius of each ball to a ny R > 2 1 − 1 /p covers t he whole space. W e show that the value 2 1 − 1 /p is optimal. 1. I N T R O DU C T I O N The so-called simultaneous packing and covering constant of a convex body C in Eucl idean space is a ce rtain m e asure of the efficiency of a packing or a covering by translates of C . This notion was introduced in var ious equivalent forms by Rogers [10], R y ˇ skov [11] an d L. F ejes T ´ oth [5], and in the lattice case c an be traced back to Delone [3]. Its study has re cently been given ren ewed attention by Zong [14, 15, 16, 17, 1 8] and others [6, 12]. Important contributions to the non -lattice c ase have also be en made by Linhardt [9], B¨ or ¨ oczky [1], an d Doyl e, Lagarias and Randall [4]. Since this n otion avoids the use of density , it can be used to study packings and coverings in hyperbolic spaces or infinite dimensional spaces. In this paper we determine the exact value of this c onstant for the ℓ p spaces where 1 ≤ p < ∞ . The main ingredient in the pr oof is an adaptation of a construction of Kl ee [7]. 2. T H E S I M U LT A N E O U S P A C KI N G A N D C O V E R I N G C O N S T A N T Let ( X , k·k ) be an y normed space. Denote the cl osed ball with center x ∈ X and r adius r by B ( x, r ) . A subset P ⊆ X is (the set of cen ters of) an r -packing if the col lection of balls { B ( x, r ) : x ∈ P } are pairwise disj oint. Equivalently , P is 2 r -disp e rsed , i.e., d ( x, y ) > 2 r for all disti nct x, y ∈ P . F or any P ⊆ X , define r ( P ) := sup { r : P is an r -packing } . A subset P ⊆ X is (the set of centers of) an R -covering (or R -net ) if the collection of balls { B ( x, R ) : x ∈ P } cover X , i.e., X = S x ∈ P B ( x, R ) . F or any P ⊆ X , define R ( P ) := in f { R : P is an R -covering } . Then R ( P ) is the supremum of the radii of balls disjoint from P : (1) R ( P ) = sup { R : for some x ∈ X , B ( x, R ) ∩ P = ∅} . If P is an r -packing, then R ( P ) − r is the supremum of the r adii of ball s that are dis joint from S p ∈ P B ( p, r ) , and if P is an R -covering , then R − r ( P ) is 1 2 KONRAD J. SW ANEPOEL the supremum of the radii of balls that are contained in more than one of B ( p, R ) , p ∈ P [5]. Definition. The simultaneous pa c king an d covering con stant of (the unit ball of) X is γ ( X ) := inf { R ( P ) : P is a 1 -packing } . W e could also have used 1 -coverings to define this constant, as shown by the identity γ ( X ) − 1 = s up { r ( P ) : P is a 1 -covering } . It is clear that R ( P ) ≥ 1 for any 1 -packing P . By Zorn’s lemma there always exists a maximal 1 -packing, which is necessarily a 2 -covering. There- fore, 1 ≤ γ ( X ) ≤ 2 . If X is finite-dimensional, then γ ( X ) is exactly the simul tane ous packing and covering constant of the u n it ball of X , as discussed in the introduction. 3. T H E M A I N T H E OR E M The main re sult of the paper conce rns the case X = ℓ p , 1 < p < ∞ , which we rec all is the space of r e al sequences x = ( x i ) i ∈ N such that P ∞ i =1 | x i | p < ∞ with nor m k x k p := ∞ X i =1 | x i | p ! 1 /p . Theorem. F or each p ∈ (1 , ∞ ) , γ ( ℓ p ) = 2 1 − 1 /p . In particular , if p is c lose to 1 , then γ ( ℓ p ) is close to 1 , which means there are very good packings of unit balls in ℓ p . P er haps more surprisingly , if p is very larg e, γ ( ℓ p ) is close to 2 , i.e., any pac king by unit balls has large holes. In the next sec tion we use a r esult of Bur lak, Rankin and Robertson [2] to show the lower bound γ ( ℓ p ) ≥ 2 1 − 1 /p . It is more difficult to fin d good pack- ings. In Sec tion 5 we adapt a construction of Kl ee [7 ] to give packings that demonstrate the upper bou n d γ ( ℓ p ) ≤ 2 1 − 1 /p . In fact, Klee already obtained this bound for ℓ p ( κ ) where κ is a regular cardinal such that κ ℵ 0 = κ . In our case, κ = ℵ 0 , and then his con struction has to be modified substantially . 4. T H E L O W E R B O U N D F or a proof of the following packing proper ty of ℓ p , see [2], [8], or [13]. Lemma 1. If the unit ball of ℓ p contains an infinite α -dispe rsed set, then α ≤ 2 1 /p . T o prove γ ( ℓ p ) ≥ 2 1 − 1 /p , it is sufficient to show the following. Proposition 2. Let P be a 1 -dispe rsed subset of ℓ p where 1 ≤ p < ∞ . Then R ( P ) ≥ 2 − 1 /p . Proof . Let 0 < ε < R ( P ) . Set r := R ( P ) − ε and δ := (( r + 2 ε ) p − r p ) 1 /p . By (1) there exists c ∈ ℓ p such that B ( c, r ) ∩ P = ∅ . T ranslate P by − c so that we may assume without loss of g enerality that c = o . Thus k x k > r for all x ∈ P . SIMUL T ANEOUS P ACKING AND COVERING IN SEQUENCE SP ACES 3 W e cl aim that Q := B ( o, r + δ + 2 ε ) ∩ P is infinite. Su ppose to the c ontrary that Q is finite. As u sual , we den ote by e n the seque nce which is 1 in position n and 0 in all other positi ons. F or any n ∈ N and x ∈ Q , k x − δ e n k p p = k x k p p − | x n | p + | x n − δ | p . Therefore, lim n →∞ k x − δ e n k p p = k x k p p + δ p > r p + δ p = ( r + 2 ε ) p . Since Q is finite, there exists n ∈ N such that for all x ∈ Q , k x − δe n k > r + 2 ε . On the other hand, for an y x ∈ P \ Q , k x k > r + δ + 2 ε , and then the triang le inequality gives k x − δ e n k > r + 2 ε . Therefore, B ( δ e n , r + 2 ε ) ∩ P = ∅ , which gives R ( P ) ≥ r + 2 ε , a contradiction. Thus Q is infinite, an d by Le m ma 1, R ( P ) + δ + ε = r + δ + 2 ε ≥ 2 − 1 /p . By l e tting ε → 0 , we obtain R ( P ) ≥ 2 − 1 /p , as required.  5. C ON S T R U C T I N G A N O P T I M A L PA C K I N G T o pr ove the u ppe r boun d γ ( ℓ p ) ≤ 2 1 − 1 /p , it is sufficient to show the following. Proposition 3. F or any 1 ≤ p < ∞ there exists a 2 1 /p -dispersed se t P ⊆ ℓ p such that R ( P ) = 1 . Proof . W e recursively construct the set P together wit h the space, which in the en d is isometric to ℓ p . If A is any set, we denote by ℓ p ( A ) the normed space of all real-val u ed functions f on A with countable support su pp( f ) := { a ∈ A : f ( a ) 6 = 0 } such that P a ∈ supp( f ) | f ( a ) | p < ∞ , and with nor m k f k p :=  X a ∈ supp( f ) | f ( a ) | p  1 /p . Thus ℓ p = ℓ p ( N ) is isometric to ℓ p ( A ) if A is countably infinite. F or any a ∈ A , let e a be the function on A such that e a ( a ) = 1 and e a ( b ) = 0 for all b ∈ A , b 6 = a . If A ⊆ A ′ , then we consider ℓ p ( A ) to be a subspace of ℓ p ( A ′ ) in the natural way . W e construct two sequenc es of countable sets P n and D n . Let P 1 = ∅ and D 1 = { 0 } = ℓ p ( ∅ ) . If P 1 , . . . , P n and D 1 , . . . , D n have been c onstructed for some n ≥ 1 , let P n +1 := { x + e x : x ∈ D n } ⊆ ℓ p  n [ i =1 D i  , and le t D n +1 be a countable dense subset of ℓ p  n [ i =1 D i  \ [ n B ( x, 1) : x ∈ n +1 [ i =1 P i o . By the definiti on of P n +1 it follows that D k ⊆ S x ∈ P k +1 B ( x, 1) for each k = 1 , . . . , n , hence D n +1 is disjoint from S n i =1 D i . It follows that the P n are also pairwise disjoint. 4 KONRAD J. SW ANEPOEL Let P := S n ∈ N P n . Then P is a subset of the space ℓ p ( S n ∈ N D n ) , which is isometric to ℓ p (note that already D 2 is infinite). W e now show that P is 2 1 /p -dispersed and is a (1 + ε ) -covering for all ε > 0 . Choose two arbitrary el e ments x + e x , y + e y ∈ P , where x ∈ D n and y ∈ D m , x 6 = y , and n ≤ m . Since supp( x ) , su pp( y ) ⊆ S m − 1 i =1 D i , which is disjoint from D m , it follows that supp( x − y ) and su pp( e y ) = { y } are disjoint. W e distinguish be tween two cases. If n = m , then sup p( e x ) = { x } is also disjoint from su pp( x − y ) and supp( e y ) , hence k ( x + e x ) − ( y + e y ) k p p = k x − y + e x − e y k p p = k x − y k p p + 1 + 1 > 2 . In the second case, n < m . Sinc e y ∈ D m , and x + e x ∈ P n +1 , it follows that y / ∈ B ( x + e x , 1) , hence k x + e x − y k p p > 1 . Since supp( x + e x − y ) an d supp( e y ) are now disjoint, k ( x + e x ) − ( y + e y ) k p p = 1 + k x + e x − y k p p > 1 + 1 . It foll ows that P is 2 1 /p -dispersed. Let ε > 0 and choose an arbitrary x ∈ ℓ p ( S n ∈ N D n ) . Choose N ∈ N large en ough such that k x − y k p < ε/ 2 for some y ∈ ℓ p ( S N − 1 i =1 D i ) . If y ∈ S { B ( z , 1) : z ∈ S N i =1 P i } , then for some z ∈ S N i =1 P i , k x − z k p ≤ k x − y k p + k y − z k p < 1 + ε/ 2 . If on the other hand y / ∈ S { B ( z , 1) : z ∈ S N i =1 P i } , then there exists z ∈ D N such that k y − z k p p < (1 + ε/ 2) p − 1 . Note that z + e z ∈ P N +1 . 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(N.S.) 39 (2002), 533– 555. 15. C. Zong, Simultaneous packing and covering in the Euclidean pl ane , Monatsh. Math. 134 (2002), 2 47–255. 16. C. Zong, Simultaneous packing and covering o f centrall y s ymmetric convex b o dies , R end. Circ. Mat. P aler mo (2) Suppl. (2002), no. 70, part II, 38 7–396. IV International Con- ference in “Stochastic Geometry , Convex Bodies, Empir ical Measur e s & Applications to Engineering S cience”, V ol. II (T ropea, 2001 ). 17. C. Zong, Simultaneous p acking and covering in three-dimension al Euclidean space , J. Lon- don Math. S oc. (2) 67 (2003 ), 29–40. 18. C. Zong, Simultaneous packi ng and covering in the Euclidean plane II , manuscr ipt, (2007 ). http://arx iv.org/ab s/0706.1808 F A K U LT ¨ A T F ¨ U R M A T H E M A T I K , T E C H N I S C H E U N I V ER S I T ¨ A T C H E M N I T Z , D - 0 9 1 0 7 C H E M N I T Z , G E R M A N Y E-mail addres s : konrad.swanepoel@ gmail.com