Macroscopic dimension of the $ell^p$-ball with respect to the $ell^q$-norm

MA CR OSCOPIC DIMENSION OF THE ℓ p -BALL WITH RESPECT TO THE ℓ q -NORM MASAKI TSUKAMOTO ∗ Abstract. W e show estimates of the “macro scopic dimension” of the ℓ p -ball with re- sp ect to the ℓ q -norm. 1. Introduction 1.1. Macroscopic dimension. Let ( X , d ) b e a compact metric space, Y a top o lo gical space. F or ε > 0, a con tin uous map f : X → Y is called an ε -em b edding if D iam f − 1 ( y ) ≤ ε for all y ∈ Y . F ollowing Gromo v [2, p. 321], w e define the “width dimension” Widim ε X as the minim um inte ger n suc h that there exist an n -dimensional p olyhedron P and an ε -em b edding f : X → P . When w e need to mak e the used distance d explicit, w e use the notation Widim ε ( X , d ). If we let ε → 0, then Widim ε giv es the usual cov ering dimension: lim ε → 0 Widim ε X = dim X. Widim ε X is a “macroscopic” dimension of X at the scale ≥ ε (cf. Gromov [2, p. 341]). It discards the informatio n of X “smaller than ε ”. F or example, [0 , 1] × [0 , ε ] (with the Euclidean distance) macroscopically lo oks one-dimensional ( ε < 1) : Widim ε [0 , 1] × [0 , ε ] = 1 . Using this notion, Gr omo v [2] defines “mean dimension”. And he prop osed op en prob- lems ab o ut this Widim ε (see [2, pp. 333-33 4]). In this pap er w e give (partial) answ ers to some of them. In [2, p. 333], he asks whether the simplex ∆ n − 1 := { x ∈ R n | x k ≥ 0 (1 ≤ k ≤ n ) , P n k =1 x k = 1 } satisfies (1) Widim ε ∆ n − 1 ∼ const ε n. Our main result b elow gives the answ er: If w e consider the standard Euclidean distance on ∆ n − 1 , then (1) do es not hold. Date : Nov ember 13, 2018. 2000 Mathematics Subje ct Classific ation. 46B 20. Key wor ds and phr ases. ℓ p -space, Widim, mean dimension. ∗ Suppo rted b y Grant-in-Aid for JSPS F ellows (19 · 1 530) fr o m Ja pan Society for the Pr omotion o f Science. 1 2 MASAKI TSUKAMOTO In [2, p. 333], he also asks what is Widim ε B ℓ p ( R n ) with resp ect to the ℓ q -norm, where (for 1 ≤ p ) B ℓ p ( R n ) := ( x ∈ R n | n X k =1 | x k | p ≤ 1 ) . Our main result concerns this problem. F or 1 ≤ q ≤ ∞ , let d ℓ q b e the ℓ q -distance on R n giv en b y d ℓ q ( x, y ) := n X k =1 | x k − y k | q ! 1 /q . W e wan t to know the v alue of Widim ε ( B ℓ p ( R n ) , d ℓ q ). Esp ecially w e are in terested in the b eha vior of Widim ε ( B ℓ p ( R n ) , d ℓ q ) as n → ∞ for small (but fixed) ε . When q = p , w e ha v e (from “Widim inequalit y” in [2, p. 333]) (2) Widim ε ( B ℓ p ( R n ) , d ℓ p ) = n for all ε < 1 . (F or its pro of, see G r o mo v [2, p. 333], Gournay [1, Lemma 2.5] or Tsuk amoto [7, App endix A].) More generally , if 1 ≤ q ≤ p ≤ ∞ , then d ℓ p ≤ d ℓ q and hence (3) Widim ε ( B ℓ p ( R n ) , d ℓ q ) = n for all ε < 1 . I think this is a satisfactory answ er. (F or the case of ε ≥ 1, there are still problems; see Gournay [1].) So the problem is the case of 1 ≤ p < q ≤ ∞ . Our main result is the follo wing: Theorem 1.1. L et 1 ≤ p < q ≤ ∞ ( q may b e ∞ ). We define r ( ≥ p ) by 1 /p − 1 /q = 1 /r . F or any ε > 0 a n d n ≥ 1 , we have (4) Widim ε ( B ℓ p ( R n ) , d ℓ q ) ≤ min( n, ⌈ (2 /ε ) r ⌉ − 1) , wher e ⌈ (2 /ε ) r ⌉ denotes the s m al lest inte ger ≥ (2 / ε ) r . Note that the right-hand-sid e of (4) b e c omes c on stant for lar ge n (a n d fixe d ε ). Ther efor e Widim ε ( B ℓ p ( R n ) , d ℓ q ) b e c o m es stable as n → ∞ . This result mak es a sharp contrast with the ab ov e (3). F or the simplex ∆ n − 1 ⊂ R n w e ha v e Widim ε ∆ n − 1 ≤ Widim ε ( B ℓ 1 ( R n ) , d ℓ 2 ) ≤ ⌈ (2 /ε ) 2 ⌉ − 1 . Therefore (1) do es no t hold. Actually this result means that the “macroscopic dimension” of ∆ n − 1 b ecomes constan t for large n . When q = ∞ , w e can prov e that the inequality (4) actually b ecomes an equalit y: Corollary 1.2. F or 1 ≤ p < ∞ , Widim ε ( B ℓ p ( R n ) , d ℓ ∞ ) = min ( n, ⌈ (2 /ε ) p ⌉ − 1) . MAC ROSCOPIC DIMENS ION O F THE ℓ p -BALL 3 This result was already obtained by A. Gourna y [1, Prop osition 1 .3]; see Remark 1.6 at the end of the intro duction. F or general q > p , I don’t ha v e an exact form ula. But w e can prov e the follow ing asymptotic result as a corollary of Theorem 1.1. Corollary 1.3. F or 1 ≤ p < q ≤ ∞ , lim ε → 0  lim n →∞ log Widim ε ( B ℓ p ( R n ) , d ℓ q ) / | log ε |  = r = pq q − p . Note that the li m it lim n →∞ log Widim ε ( B ℓ p ( R n ) , d ℓ q ) exi s ts b e c ause Widim ε ( B ℓ p ( R n ) , d ℓ q ) is m onotone non-de cr e a s ing in n and has an upp er b ound ind ep endent of n . Remark 1.4. Gournay [1, Example 3.1] sho ws Widim ε ( B ℓ 1 ( R 2 ) , d ℓ p ) = 2 for ε < 2 1 /p . 1.2. Mean dimension t heory . Theorem 1.1 has an application to Gr omo v’s mean di- mension theory . Let Γ b e a infinite coun table group. F or 1 ≤ p ≤ ∞ , let ℓ p (Γ) ⊂ R Γ b e the ℓ p -space, B ( ℓ p (Γ)) ⊂ ℓ p (Γ) the unit ball (in the ℓ p -norm). W e consider the na t ur a l righ t action of Γ on ℓ p (Γ) (and B ( ℓ p (Γ))): ( x · δ ) γ := x δγ for x = ( x γ ) γ ∈ Γ ∈ ℓ p (Γ) and δ ∈ Γ . W e give the standard pro duct top olo gy o n R Γ , and consider t he restriction of this top olo g y to B ( ℓ p (Γ)) ⊂ R Γ . (This top olog y coincides with the r estriction of w eak top ology of ℓ p (Γ) for p > 1.) Then B ( ℓ p (Γ)) b ecomes compact and metrizable. (The Γ- action on B ( ℓ p (Γ)) is con tin uous.) Let d b e t he distance o n B ( ℓ p (Γ)) compatible with the top o lo gy . F o r a finite subset Ω ⊂ Γ w e define a distance d Ω on B ( ℓ p (Γ)) b y d Ω ( x, y ) := max γ ∈ Ω d ( xγ , y γ ) . W e are in terested in the grow th b eha vior of Widim ε ( B ( ℓ p (Γ)) , d Ω ) as | Ω | → ∞ . In particular, if Γ is finitely generated and has an amenable sequenc e { Ω i } i ≥ 1 (in the sense of [2, p. 335]), the mean dimension is defined by (see [2, pp. 335-33 6]) dim( B ( ℓ p (Γ)) : Γ) = lim ε → 0 lim i →∞ Widim ε ( B ( ℓ p (Γ)) , d Ω i ) / | Ω i | . As a corollary of Theorem 1.1, w e get the fo llowing: Corollary 1.5. F or 1 ≤ p < ∞ and any ε > 0 , ther e is a p ositive c o n stant C ( d, p, ε ) < ∞ (indep endent of Ω ) such that (5) Widim ε ( B ( ℓ p (Γ)) , d Ω ) ≤ C ( d, p, ε ) for al l finite set Ω ⊂ Γ . Namely, Widim ε ( B ( ℓ p (Γ)) , d Ω ) b e c omes stable for la r g e Ω ⊂ Γ . I n p a rticular, for a finitely gener ate d infi n ite amenable gr oup Γ (6) dim( B ( ℓ p (Γ)) : Γ ) = 0 . (6) is the answ er to the question of G romo v in [2, p. 340]. Actually t he ab ov e (5) is m uch stronger than (6). 4 MASAKI TSUKAMOTO Remark 1.6. This pap er is a revised v ersion o f the preprint [5]. A referee of [5] p o inted out that the ab ov e (6) can b e deriv ed from the theorem of Lindenstrauss-W eiss [4, The- orem 4.2]. This theorem tells us tha t if the top ological en tropy is finite then the mean dimension b ecomes 0. W e can see that the to p ological en trop y of B ( ℓ p (Γ)) (under the Γ-action) is 0. Hence the mean dimension also b ecomes 0. I am most grateful to the referee of [5] for p oin ting out this argumen t. The essen tial par t of the pro of of Theo- rem 1.1 (and Corollary 1.2 and Corollary 1.3) is the construction of the con tinuous map f : R n → R n described in Section 3 . This construction w a s already g iven in the preprin t [5]. When I was writing this revised v ersion of [5], I found the pap er of A. Gournay [1]. [1] prov es Corollary 1.2 ([1, Prop osition 1.3]) b y using essen tially the same contin uous map as men tioned ab ov e. I submitted the pap er [5] to a certain journal in June of 2007 b efore [1] app eared on the arXiv in No v ember of 20 0 7. And [5] is quoted as one of the references in [1]. 2. pr eliminaries Lemma 2.1. F or s ≥ 1 and x, y , z ≥ 0 , if x ≥ y , then x s + ( y + z ) s ≤ ( x + z ) s + y s . Pr o of. Set ϕ ( t ) := ( t + z ) s − t s ( t ≥ 0). Then ϕ ′ ( t ) = s { ( t + z ) s − 1 − t s − 1 } ≥ 0. Hence ϕ ( y ) ≤ ϕ ( x ), i.e., ( y + z ) s − y s ≤ ( x + z ) s − x s .  Lemma 2.2. L et s ≥ 1 and c, t ≥ 0 . If r e al numb ers x 1 , · · · , x n ( n ≥ 1 ) sa tisfies x 1 + · · · + x n ≤ c, 0 ≤ x i ≤ t (1 ≤ i ≤ n ) , then x s 1 + · · · + x s n ≤ c · t s − 1 . Pr o of. First we suppo se nt ≤ c . Then x s 1 + · · · + x s n ≤ n · t s ≤ c · t s − 1 . Next w e suppo se nt > c . Let m := ⌊ c/t ⌋ b e the maxim um inte ger satisfying mt ≤ c . W e ha v e 0 ≤ m < n and c − mt < t . Using Lemma 2.1, w e ha v e x s 1 + · · · + x s n ≤ t s + · · · + t s | {z } m +( c − mt ) s ≤ mt s + t s − 1 ( c − mt ) ≤ c · t s − 1 .  3. Proof of Theorem 1.1 Let S n b e the n -th symmetric group. W e define the gr o up G b y G := {± 1 } n ⋊ S n . The multiplication in G is given b y (( ε 1 , · · · , ε n ) , σ ) · (( ε ′ 1 , · · · , ε ′ n ) , σ ′ ) := (( ε 1 ε ′ σ − 1 (1) , · · · , ε n ε ′ σ − 1 ( n ) ) , σ σ ′ ) MAC ROSCOPIC DIMENS ION O F THE ℓ p -BALL 5 where ε 1 , · · · , ε n , ε ′ 1 · · · , ε ′ n ∈ { ± 1 } and σ, σ ′ ∈ S n . G acts on R n b y (( ε 1 , · · · , ε n ) , σ ) · ( x 1 , · · · , x n ) := ( ε 1 x σ − 1 (1) , · · · , ε n x σ − 1 ( n ) ) where (( ε 1 , · · · , ε n ) , σ ) ∈ G and ( x 1 , · · · , x n ) ∈ R n . The a ction of G on R n preserv es the ℓ p -ball B ℓ p ( R n ) and the ℓ q -distance d ℓ q ( · , · ). W e define R n ≥ 0 and Λ n b y R n ≥ 0 := { ( x 1 , · · · , x n ) ∈ R n | x i ≥ 0 (1 ≤ i ≤ n ) } , Λ n := { ( x 1 , · · · , x n ) ∈ R n | x 1 ≥ x 2 ≥ · · · ≥ x n ≥ 0 } . The f ollo wing can b e easily c heck ed: Lemma 3.1. F or ε ∈ {± 1 } n and x ∈ R n ≥ 0 , if εx ∈ R n ≥ 0 , then εx = x . F o r σ ∈ S n and x ∈ Λ n , if σ x ∈ Λ n , then σ x = x . F or g = ( ε, σ ) ∈ G and x ∈ Λ n , if g x ∈ Λ n , then g x = ε ( σ x ) = σ x = x . Let m, n b e p ositiv e inte gers suc h that 1 ≤ m < n . W e define the con tinuous map f 0 : Λ n → Λ n b y f 0 ( x 1 , · · · , x n ) := ( x 1 − x m +1 , x 2 − x m +1 , · · · , x m − x m +1 , 0 , 0 , · · · , 0 | {z } n − m ) . The f ollo wing is the key fact for o ur construction: Lemma 3.2. F or g ∈ G a n d x ∈ Λ n , if g x ∈ Λ n ( ⇒ g x = x ), then we have f 0 ( g x ) = g f 0 ( x ) . Pr o of. First we consider the case of g = ε = ( ε 1 , · · · , ε n ) ∈ {± 1 } n . If x m +1 = 0, then f 0 ( εx ) = ( ε 1 x 1 , · · · , ε m x m , 0 , · · · , 0) = εf 0 ( x ) . If x m +1 > 0, then ε i = 1 (1 ≤ i ≤ m + 1) b ecause ε i x i = x i ≥ x m +1 > 0 (1 ≤ i ≤ m + 1 ). Hence f 0 ( εx ) = ( x 1 − x m +1 , · · · , x m − x m +1 , 0 , · · · , 0) = f 0 ( x ) = εf 0 ( x ) . Next w e consider the case of g = σ ∈ S n . g x ∈ Λ n implies x σ ( i ) = x i (1 ≤ i ≤ n ). Set y := f 0 ( x ). Let r (1 ≤ r ≤ m + 1) b e the integer suc h that x r − 1 > x r = x r +1 = · · · = x m +1 . F rom x σ ( i ) = x i (1 ≤ i ≤ n ), w e ha v e 1 ≤ i < r ⇒ 1 ≤ σ ( i ) < r ⇒ y σ ( i ) = x σ ( i ) − x m +1 = y i , r ≤ i ⇒ r ≤ σ ( i ) ⇒ y σ ( i ) = 0 = y i . Hence we ha v e f 0 ( σ x ) = f 0 ( x ) = σ f 0 ( x ). 6 MASAKI TSUKAMOTO Finally we consider the case of g = ( ε, σ ) ∈ G . Since g x ∈ Λ n , we ha v e g x = ε ( σ x ) = σ x = x ∈ Λ n (see Lemma 3.1). Hence f 0 ( g x ) = f 0 ( ε ( σ x )) = εf 0 ( σ x ) = εσ f 0 ( x ) = g f 0 ( x ) .  W e define a con tin uous map f : R n → R n as follow s; F or an y x ∈ R n , there is a g ∈ G suc h that g x ∈ Λ n . Then w e define f ( x ) := g − 1 f 0 ( g x ) . F rom Lemma 3 .2, this definition is w ell-defined. Sinc e R n = S g ∈ G g Λ n and f | g Λ n = g f 0 g − 1 ( g ∈ G ) is contin uous on g Λ n , f is contin uous on R n . Moreov er f is G - equiv arian t. Prop osition 3.3. L et 1 ≤ p < q ≤ ∞ . F or any x ∈ B ℓ p ( R n ) , w e have d ℓ q ( x, f ( x )) ≤  1 m + 1  1 p − 1 q . Note that the right-hand side do es not dep end on n . Pr o of. Since f is G -equiv arian t and d ℓ q is G -in v aria n t , w e can supp ose x ∈ B ℓ p ( R n ) ∩ Λ n , i.e. x = ( x 1 , x 2 , · · · , x n ) with x 1 ≥ x 2 ≥ · · · ≥ x n ≥ 0. W e ha v e f ( x ) = ( x 1 − x m +1 , · · · , x m − x m +1 , 0 , · · · , 0) . Hence d ℓ q ( x, f ( x )) = | | ( x m +1 , · · · , x m +1 | {z } m +1 , x m +2 , · · · , x n ) | | ℓ q . Set t := x p m +1 and s := q /p . Since x p 1 + · · · + x p n ≤ 1 and x 1 ≥ x 2 ≥ · · · ≥ x n ≥ 0 , w e hav e t ≤ 1 / ( m + 1), 0 ≤ x p k ≤ t ( m + 1 ≤ k ≤ n ) and x p m +2 + · · · + x p n ≤ 1 − ( m + 1) t . Using Lemma 2.2 , w e ha v e x q m +2 + · · · + x q n ≤ { 1 − ( m + 1) t } t s − 1 = t s − 1 − ( m + 1) t s . Therefore d ℓ q ( x, f ( x )) q = ( m + 1) x q m +1 + x q m +2 + · · · + x q n ≤ t s − 1 ≤ (1 / ( m + 1)) s − 1 . Th us d ℓ q ( x, f ( x )) ≤ (1 / ( m + 1)) 1 /p − 1 /q .  Pr o of of The or em 1 . 1. Set m := min( n, ⌈ (2 /ε ) r ⌉− 1). W e will pro ve Widim ε ( B ℓ p ( R n ) , d ℓ q ) ≤ m . If n = m , then the statemen t is trivial. Henc e w e supp ose n > m = ⌈ (2 /ε ) r ⌉ − 1 . F rom m + 1 = ⌈ (2 /ε ) r ⌉ ≥ (2 /ε ) r and 1 /r = 1 / p − 1 / q , 2  1 m + 1  1 p − 1 q ≤ ε. MAC ROSCOPIC DIMENS ION O F THE ℓ p -BALL 7 W e ha v e f ( R n ) = [ g ∈ G g f ( Λ n ) . Note tha t f ( Λ n ) ⊂ R m := { ( x 1 , · · · , x m , 0 , · · · , 0) ∈ R n } . Prop o sition 3.3 implies that f | B ℓ p ( R n ) : ( B ℓ p ( R n ) , d ℓ q ) → [ g ∈ G g · R m is a 2  1 m + 1  1 p − 1 q -em b edding . Therefore w e get Widim ε ( B ℓ p ( R n ) , d ℓ q ) ≤ m .  4. Proof of Cor ollaries 1.2 and 1.3 4.1. P ro of of Corollary 1.2. W e need the follow ing result. ( cf. Gromo v [2, p. 33 2]. F or its pro of, see Lindenstrauss-W eiss [4, Lemma 3.2] or Tsuk amoto [6, Example 4.1].) Lemma 4.1. F or ε < 1 , Widim ε ([0 , 1] n , d ℓ ∞ ) = n, wher e d ℓ ∞ is the sup-distanc e given by d ℓ ∞ ( x, y ) := max i | x i − y i | .@ F rom this we get: Lemma 4.2. L et B ℓ ∞ ( R n , ρ ) b e the close d b a l l of r a dius ρ c enter e d a t the origin in ℓ ∞ ( R n ) ( ρ > 0 ). Then for ε < 2 ρ Widim ε ( B ℓ ∞ ( R n , ρ ) , d ℓ ∞ ) = n. Pr o of. Consider the bijection [0 , 1] n → B ℓ ∞ ( R n , ρ ) , ( x 1 , · · · , x n ) 7→ ( 2 ρx 1 − ρ, · · · , 2 ρx n − ρ ) . Then the statemen t easily follows from Lemma 4.1.  Pr o of of Cor ol lary 1.2. Set m := min( n, ⌈ (2 /ε ) p ⌉ − 1). W e already know (Theorem 1 .1) Widim ε ( B ℓ p ( R n ) , d ℓ ∞ ) ≤ m . W e w an t to sho w Widim ε ( B ℓ p ( R n ) , d ℓ ∞ ) ≥ m . Note t ha t f or an y real n umber x w e ha ve ⌈ x ⌉ − 1 < x . Hence m ≤ ⌈ (2 /ε ) p ⌉ − 1 < (2 / ε ) p . There fore m ( ε/ 2) p < 1. Then if we c ho ose ρ > ε / 2 sufficien t ly close to ε/ 2 , then ( m ≤ n ) B ℓ ∞ ( R m , ρ ) ⊂ B ℓ p ( R n ) . (If ε ≥ 2, then m = 0 and B ℓ ∞ ( R m , ρ ) is { 0 } .) F rom Lemma 4.2 , Widim ε ( B ℓ p ( R n ) , d ℓ ∞ ) ≥ Widim ε ( B ℓ ∞ ( R m , ρ ) , d ℓ ∞ ) = m. Essen tially t he same argument is given in G ourna y [1 , pp. 5-6 ].  8 MASAKI TSUKAMOTO 4.2. P ro of of Cor ollary 1.3. The follo wing lemma easily follows from (2) Lemma 4.3. L et B ℓ q ( R n , ρ ) b e the clos e d b al l of r ad ius ρ c enter e d at the origin in ℓ q ( R n ) ( 1 ≤ q ≤ ∞ and ρ > 0 ). F or ε < ρ , Widim ε ( B ℓ q ( R n , ρ ) , d ℓ q ) = n. Prop osition 4.4. F or 1 ≤ p < q ≤ ∞ , min( n, ⌈ ε − r ⌉ − 1) ≤ Widim ε ( B ℓ p ( R n ) , d ℓ q ) , wher e r is define d by 1 /r = 1 /p − 1 /q . Pr o of. W e can supp ose q < ∞ . Set m := min( n, ⌈ ε − r ⌉ − 1). F rom H¨ older’s inequalit y , ( | x 1 | p + · · · + | x m | p ) 1 /p ≤ m 1 /r ( | x 1 | q + · · · + | x m | q ) 1 /q . As in t he pro of o f Corollary 1 .2, w e ha v e m ≤ ⌈ ε − r ⌉ − 1 < ε − r , i.e. m 1 /r ε < 1 . Therefore if w e c ho ose ρ > ε sufficien tly close to ε , then ( m ≤ n ) B ℓ q ( R m , ρ ) ⊂ B ℓ p ( R n ) . F rom Lemma 4.3 , Widim ε ( B ℓ p ( R n ) , d ℓ q ) ≥ Widim ε ( B ℓ q ( R m , ρ ) , d ℓ q ) = m.  Pr o of of Cor ol lary 1.3. F r o m Theorem 1.1 and Prop osition 4 .4, w e hav e ⌈ ε − r ⌉ − 1 ≤ lim n →∞ Widim ε ( B ℓ p ( R n ) , d ℓ q ) ≤ ⌈ (2 /ε ) r ⌉ − 1 . F rom this estimate, we can easily get the conclusion.  5. Proof of Corollar y 1.5 Let 1 ≤ p < ∞ and ε > 0. Set X := B ( ℓ p (Γ)). T o b egin with, w e w an t to fix a distance on X (compatible with the top ology). Since X is compact, if w e pro ve (5) for one fixed distance, then (5) b ecomes v alid for any distance on X . Let w : Γ → R > 0 b e a p ositiv e function satisfying X γ ∈ Γ w ( γ ) ≤ 1 . W e define the distance d ( · , · ) on X b y d ( x, y ) := X γ ∈ Γ w ( γ ) | x γ − y γ | for x = ( x γ ) γ ∈ Γ and y = ( y γ ) γ ∈ Γ in X . As in Section 1, w e define the distance d Ω on X for a finite subset Ω ⊂ Γ by d Ω ( x, y ) := max γ ∈ Ω d ( xγ , y γ ) . MAC ROSCOPIC DIMENS ION O F THE ℓ p -BALL 9 F or each δ ∈ Γ, there is a finite set Ω δ ⊂ Γ suc h that X γ ∈ Γ \ Ω δ w ( δ − 1 γ ) ≤ ε/ 4 . Set Ω ′ := S δ ∈ Ω Ω δ . Ω ′ is a finite set satisfying X γ ∈ Γ \ Ω ′ w ( δ − 1 γ ) ≤ ε/ 4 for an y δ ∈ Ω . Set c := ⌈ (4 /ε ) p ⌉ − 1. Let π : X → B ℓ p ( R Ω ′ ) = { x ∈ R Ω ′ | | | x | | p ≤ 1 } b e the natural pro jection. F rom Theorem 1.1, there are a p olyhedron K of dimension ≤ c a nd an ε/ 2- em b edding f : ( B ℓ p ( R Ω ′ ) , d ℓ ∞ ) → K . Then F := f ◦ π : ( X , d Ω ) → K b ecomes an ε -em b edding; If F ( x ) = F ( y ), then d ℓ ∞ ( π ( x ) , π ( y )) ≤ ε / 2 and for each δ ∈ Ω d ( xδ, y δ ) = X γ ∈ Ω ′ w ( δ − 1 γ ) | x γ − y γ | + X γ ∈ Γ \ Ω ′ w ( δ − 1 γ ) | x γ − y γ | , ≤ ε 2 X γ ∈ Ω ′ w ( δ − 1 γ ) + 2 X γ ∈ Γ \ Ω ′ w ( δ − 1 γ ) , ≤ ε/ 2 + ε/ 2 = ε. Hence d Ω ( x, y ) ≤ ε . Therefore, Widim ε ( X , d Ω ) ≤ c. This shows (5). If Γ has a n amenable sequence { Ω i } i ≥ 1 , then | Ω i | → ∞ and hence lim i →∞ Widim ε ( X , d Ω i ) / | Ω i | = 0 . This shows (6). Reference s [1] A. Gour nay , Widths of l p balls, arXiv:0 711.30 81 [2] M. Gromov, T opolo gical in v aria nt s o f dynamical systems and space s o f holomorphic maps: I, Math. Phys. Anal. Ge o m. 2 (1 9 99) 32 3-415 [3] E. Lindenstra uss, Mean dimension, small ent ropy factors and an embedding theorem, Inst. Hautes ´ Etudes Sci. Publ. Math. 89 (1999 ) 22 7 -262 [4] E. Lindenstra uss, B . W eiss, Mean top o logical dimensio n, Israel J. Ma th. 1 15 (20 00) 1- 24 [5] M. Tsuk amoto, Mean dimension of the unit ball in ℓ p , preprint, ht tp://www.math.kyoto-u.ac.jp/pr eprint/index.h tml, (2007) [6] M. Tsuk a moto, Mo duli space of Bro dy curves, energy and mean dimension, preprint, ar Xiv:0706 .2981 [7] M. Tsuk amo to , Deforma tion of Br o dy c ur ves a nd mean dimensio n, prepr int , a rXiv:071 2.0266 Masaki Tsuk amoto Departmen t o f Mathematics, F a cult y of Science Ky o t o Univ ersit y 10 MASAKI TSUKAMOTO Ky o t o 606- 8502 Japan E-mail addr e s s : tukamoto@math.kyoto- u.ac.jp