aa r X i v : . [ m a t h . M G ] F e b Widths of l p balls Antoine GournayNovember 2, 2007
Let ( X , d ) be a metric space and e ∈ R > , then we say a map f : X → Y is an e -embedding if it is continuous and the diameter of the fibres is less than e , i.e. ∀ y ∈ Y , Diam f − ( y ) ≤ e . We will use the notation f : X e ֒ → Y . This type of maps, which canbe traced at least to the work of Pontryagin (see [13] or [8]), is related to the notion ofUrysohn width (sometimes referred to as Alexandrov width), a n ( X ) , see [3]. It is thesmallest real number such that there exists an e -embedding from X to a n -dimensionalpolyhedron. Surprisingly few estimations of these numbers can be found, and one ofthe aims of this paper is to present some. However, following [7], we shall introduce: Definition 1.1: wdim e X is the smallest integer k such that there exists an e -embedding f : X → K where K is a k-dimensional polyhedron.wdim e ( X , d ) = inf X e ֒ → K dim K . Thus, it is equivalent to be given all the Urysohn’s widths or the whole data ofwdim e X as a function of e . Definition 1.2:
The wdim spectrum of a metric space ( X , d ) , denoted wspec X ⊂ Z ≥ ∪{ + ¥ } , is the set of values taken by the map e wdim e X .The a n ( X ) obviously form an non-increasing sequence, and the points of wspec X are precisely the integers for which it decreases. We shall be interested in the widthsof the following metric spaces: let B l p ( n ) be the set given by the unit ball in R n for the l p metric ( k ( x i ) k l p = (cid:0) (cid:229) | x i | p (cid:1) / p ), but look at B l p ( n ) with the l ¥ metric ( i.e. the supmetric of the product). Then Proposition 1.3: wspec ( B l p ( n ) , l ¥ ) = { , , . . ., n } ,and, ∀ e ∈ R > ,wdim e ( B l p ( n ) , l ¥ ) = ≤ e k if 2 ( k + ) − / p ≤ e < k − / p n if e < n − / p . The important outcome of this theorem is that for fixed e , the wdim e ( B l p ( n ) , l ¥ ) is bounded from below by min ( n , m ( p , e )) and from above by min ( n , M ( p , e )) , where m , M are independent of n . As an upshot high values can only be reached for small e independantly of n . It can be used to show that the mean dimension of the unit ball of1 p ( G ) , for G a countable group, with the natural action of G and the weak- ∗ topology iszero when p < ¥ (see [14]). It is one of the possible ways of proving the non-existenceof action preserving homeomorphisms between l ¥ ( G ) and l p ( G ) ; a simpler argumentwould be to notice that with the weak- ∗ topology, G sends all points of l p ( G ) to 0 while l ¥ ( G ) has many periodic orbits.The behaviour is quite different when balls are looked upon with their natural met-ric. Theorem 1.4:
Let p ∈ [ , ¥ ) , n >
1, then ∃ h n ∈ Z satisfying h n = n / n even, h = h n = n + or n − otherwise,suchthat { , h ( n ) , n }∪ ⊂ wspec ( B l p ( n ) , l p ) ⊂ { } ∪ ( n − , n ] ∩ Z . When p = p = n +
1, then n − ( B l p ( n ) , l p ) .Moreprecisely,let k , n ∈ N with n − < k < n . Thenthereexists b n ; p ∈ [ , ] and c k , n ; p ∈ [ , ) suchthatif e ≥ e ( B l p ( n ) , l p ) = e < e ( B l p ( n ) , l p ) > n − e ≥ c k , n ; p then wdim e ( B l p ( n ) , l p ) ≤ k if e < b k ; p then wdim e ( B l p ( n ) , l p ) ≥ k and, for fixed n and p , the sequence c k , n ; p is non-increasing. Furthermore, b k ; p ≥ / p ′ (cid:0) + k (cid:1) / p when 1 ≤ p ≤ b k ; p ≥ / p (cid:0) + k (cid:1) / p ′ if 2 ≤ p < ¥ .Additionally,intheEuclideancase( p = b n ;2 = c n − , n ;2 = q ( + n ) ,whileinthe2-dimensionalcase b p ≥ max ( / p , / p ′ ) forany p ∈ [ , ¥ ] . Also,if p =
1, and there is a Hadamardmatrix in dimension n + b n ;1 = c n − , n ;1 = (cid:0) + n (cid:1) .Finally, when n = ∀ e > , wdim e B l p ( n ) = c , p ≤ ( ) / p , which means inparticularthat c , p = b p when p ∈ [ , ] .Various techniques are involved to achieve this result; they will be exposed in sec-tion 3. While upper bounds on wdim e X are obtained by writing down explicit maps toa space of the proper dimension (these constructions use Hadamard matrices), lowerbounds are found as consequences of the Borsuk-Ulam theorem, the filling radius ofspheres, and lower bounds for the diameter of sets of n + ≤ p ≤ wdim e Here are a few well established results; they can be found in [1], [2], [11], and [12].
Proposition 2.1:
Let ( X , d ) and ( X ′ , d ′ ) betwometricspaces. wdim e hasthefollowingproperties:a. If X admitsatriangulation,wdim e ( X , d ) ≤ dim X .2. Thefunction e wdim e X isnon-increasing.c. Let X i betheconnectedcomponentsof X ,thenwdim e ( X , d ) = ⇔ e ≥ max i Diam X i .d. If f : ( X , d ) → ( X ′ , d ′ ) isacontinuousfunctionsuchthat d ( x , x ) ≤ Cd ′ ( f ( x ) , f ( x )) where C ∈ ] , ¥ [ ,then wdim e ( X , d ) ≤ wdim e / C ( X ′ , d ′ ) .e. Dilationsbehaveasexpected,i.e. let f : ( X , d ) → ( X ′ , d ′ ) beanhomeomorphismsuchthat d ( x , x ) = Cd ′ ( f ( x ) , f ( x )) ;thisequalitypassesthroughtothewdim:wdim e ( X , d ) = wdim e / C ( X ′ , d ′ ) .f. If X iscompact,then ∀ e > , wdim e ( X , d ) < ¥ . Proof.
They are brought forth by the following remarks:a. If dim X = ¥ , the statement is trivial. For X a finite-dimensional space, it sufficesto look at the identity map from X to its triangulation T ( X ) , which is continuousand injective, thus an e -embedding ∀ e .b. If e ≤ e ′ , an e -embedding is also an e ′ -embedding.c. If wdim e X = ∃ f : X e ֒ → K where K is a totally discontinuous space. ∀ k ∈ K , f − ( k ) is both open and closed, which implies that it contains at least oneconnected component, consequently Diam X i ≤ e . On the other hand, if e ≥ Diam X i the map that sends every X i to a point is an e -embedding.d. If wdim e / C X ′ = n , there exists an e C -embedding f : X ′ → K with dim K = n .Noticing that the map f ◦ f is an e -embedding from X to K allows us to sustainthe claimed inequality.e. This statement is a simple application of the preceding for f and f − .f. To show that wdim e is finite, we will use the nerve of a covering; see [8, §V.9]for example. Given a covering of X by balls of radius less than e /
2, there exists,by compactness, a finite subcovering. Thus, sending X to the nerve of this finitecovering is an e -immersion in a finite dimensional polyhedron.Another property worth noticing is that lim e → wdim e ( X , d ) = dim X for compact X ;we refer the reader to [1, prop 4.5.1]. Reading [6, app.1] leads to believe that there isa strong relation between wdim and the quantities defined therein (Rad k and Diam k );the existence of a relation between wdim and the filling radius becomes a natural idea,implicit in [7, 1.1B]. We shall make a small parenthesis to remind the reader of thedefinition of this concept, it is advised to look in [6, §1] for a detailed discussion.Let ( X , d ) be a compact metric space of dimension n , and let L ¥ ( X ) be the (Banach)space of real-valued bounded functions on X , with the norm k f k L ¥ = sup x ∈ X | f ( x ) | . Themetric on X yields an isometric embedding of X in L ¥ ( X ) , known as the Kuratowskiembedding: I X : X → L ¥ ( X ) x f x ( x ′ ) = d ( x , x ′ ) . k f x − f x ′ k L ¥ = sup x ′′ ∈ X (cid:12)(cid:12) d ( x , x ′′ ) − d ( x ′ , x ′′ ) (cid:12)(cid:12) = d ( x , x ′ ) . Denote by U e ( X ) the neighborhood of X ⊂ L ¥ ( X ) given by all points at distance lessthan e from X , i.e. U e ( X ) = (cid:8) f ∈ L ¥ ( X ) (cid:12)(cid:12) inf x ∈ X k f − f x k L ¥ < e (cid:9) . Definition 2.2:
The filling radius of a n -dimensional compact metric space X , writtenFilRad X , is defined as the smallest e such that X bounds in U e ( X ) , i.e. I X ( X ) ⊂ U e ( X ) induces a trivial homomorphism in simplicial homology H n ( X ) → H n ( U e ( X )) .Though FilRad can be defined for an arbitrary embedding, we will only be con-cerned with the Kuratowski embedding. Lemma 2.3:
Let ( X , d ) bea n -dimensionalcompactmetricspace, k < n aninteger,and Y ⊂ X a k -dimensionalclosed set representingatrivial(simplicial)homologyclass in H k ( X ) . Then e < Y ⇒ wdim e ( X , d ) > k . If we remove the assumption that [ Y ] ∈ H k ( X ) be trivial, the inequality is no longerstrict: wdim e ( X , d ) ≥ k . Proof.
Let us show that wdim e ( X , d ) ≤ k ⇒ e ≥ Y . Given an e -embedding f : X e ֒ → K , then f ( Y ) ⊂ K bounds, since f ∗ [ Y ] = [ Y ] = H k ( X ) . Since dim K ≤ k = dim Y , the chain representing f ( Y ) is trivial. Compactness of X allows us to supposethat f is onto a compact K . Otherwise, we restrict the target to f ( X ) . We will nowproduce a map Y → L ¥ ( Y ) whose image is contained in U e / ( Y ) , so that Y will boundin its e -neighborhood. This will mean that e ≥ Y . Let Q : K → L ¥ ( X ) k g k ( x ′′ ) = e / + inf x ′ ∈ f − ( k ) d ( x ′′ , x ′ ) , and r Y : L ¥ ( X ) → L ¥ ( Y ) f f | Y . First, notice that r Y ◦ Q ◦ f ( Y ) ⊂ U d + e / ( Y ) , ∀ d > k r Y ◦ Q ◦ f ( y ) − I Y ( y ) k L ¥ = sup y ′′ ∈ Y (cid:12)(cid:12)(cid:12)(cid:12) e + (cid:20) inf y ′ ∈ f − ( f ( y )) d ( y ′′ , y ′ ) (cid:21) − d ( y ′′ , y ) (cid:12)(cid:12)(cid:12)(cid:12) = e / , since f is an e -embedding. Second, ( r Y ◦ Q ◦ f ) ∗ [ Y ] = ( r Y ◦ Q ◦ f ) ∼ I Y in U d + e / ( Y ) , as L ¥ ( Y ) is a vector space. Consequently, [ I Y ( Y )] = e ≥ Y ,by letting d → [ Y ] = ⊂ H k ( X ) , the proof still follows by taking K of dimension k −
1: thehomology class f ∗ [ Y ] is then inevitably trivial, since K has no rank k homology.Thus, calculating FilRad is a good starting point. The following lemma consists ofa lower bound for FilRad: Lemma 2.4:
Let X be a closed convex set in a n -dimensional normed vector space.Suppose it contains a point x such that the convex hull of n + ¶ X whosediameter is < a excludes x . Then FilRad ¶ X ≥ a /
2, and, using lemma 2.3, e < a ⇒ wdim e X = n . 4 roof. Suppose that Y = ¶ X has a filling radius less than a /
2. Then, ∃ e > ∃ P apolyhedron such that Y bounds in P , P ⊂ U a − e ( Y ) and that the simplices of P have adiameter less than e . To any vertex p ∈ P it is possible to associate f ( p ) ∈ I Y ( Y ) so that k p , f ( p ) k L ¥ ( Y ) < a − e . Let p , . . . , p n be a n -simplex of P ,Diam { f ( p ) , . . . , f ( p n ) } < ( a − e ) + e < a − e < a . Since I Y is an isometry, f ( p i ) can be seen as points of Y without changing the diameterof the set they form. The convex hull of these f ( p i ) in B will not contain x : theirdistance to f ( p ) is < a which excludes x . Let p be the projection away from x , thatis associate to x ∈ X , the point p ( x ) ∈ ¶ X on the half-line joining x to x . Using p ,the n -simplex generated by the f ( p i ) yields a simplex in Y . Thus we extended f to aretraction r from P to Y . Let c be a n -chain of P which bounds Y , i.e. [ Y ] = d c . Acontradiction becomes apparent: [ Y ] = r ∗ [ Y ] = r ∗ d c = d r ∗ c . Indeed, if that was to betrue, Y , which is n − n -dimensional chain in Y .Hence FilRad Y > a / Lemma 2.5: (cf. [7, 1.1B]) Let B be the unit ball of a n -dimensionalBanach space,then ∀ e < , wdim e B = n . Proof.
Any set of n + Y = ¶ B whose diameter is less than 1 does not containthe origin in its convex hull. So according to lemma 2.4, FilRad Y > /
2, and since Y is a closed set of dimension n − B , we conclude byapplying lemma 2.3.Let us emphasise this important fact on l ¥ balls in finite dimensional space. Lemma 2.6:
Let B l ¥ ( n ) = [ − , ] n betheunitcubeof R n withtheproduct(supremum)metric,then wdim e B l ¥ ( n ) = (cid:26) e ≥ n if e < . This lemma will be used in the proof of proposition 1.3. Its proof, which uses theBrouwer fixed point theorem and the Lebesgue lemma, can be found in [12, lem 3.2],[2, prop 2.7] or [1, prop 4.5.4].
Proof of proposition 1.3:
We first show the lower bound on wdim e . In a k -dimensionalspace, the l ¥ ball of radius k − / p is included in the l p ball: B l ¥ ( k ) k − / p ⊂ B l p ( k ) , as k x k l p ( k ) ≤ k / p k x k l ¥ ( k ) . Since B l p ( k ) ⊂ B l p ( n ) , by 2.1.d, we are assured that, if B l p ( n ) is consideredwith the l ¥ metric, e < k − / p implies that wdim e ( B l p ( n ) , l ¥ ) ≥ k .To get the upper bound, we give explicit e -embeddings to finite dimensional poly-hedra. This will be done by projecting onto the union of ( n − j ) -dimensional coordi-nates hyperplanes (whose points have at least j coordinates equal to 0). Project a point x ∈ B l p ( n ) by the map p j as follows: let m be its j th smallest coordinate (in absolutevalue), set it and all the smaller coordinates to 0, other coordinates are substracted m ifthey are positive or added m if they are negative.5enote by ~ e an element of {− , } n and ~ e \ A the same vector in which ∀ i ∈ A , e i isreplaced by 0. The largest fibre of the map p j is p − j ( ) = ∪ ~ e , i ,..., i j − { l ~ e + (cid:229) ≤ l ≤ j − l l ~ e \{ i ,..., i l } | l i ∈ R ≥ } ∩ B l p ( n ) . Its diameter is achieved by s = (cid:0) ( n − j + ) − / p , . . . , ( n − j + ) − / p , , . . . , (cid:1) and − s ;thus Diam p − j ( ) = ( n − j + ) − / p . p j allows us to assert that e > ( n − j + ) − / p ⇒ wdim e ( B l p ( n ) , l ¥ ) ≤ n − j , by realising a continuous map in a ( n − j ) -dimensional poly-hedron whose fibres are of diameter less than 2 ( n − j + ) − / p .The above proof for an upper bound also gives that wdim e ( B l p ( n ) , l q ) ≤ k if e ≥ ( k + ) / q − / p , but the inclusion of a l q ball of proper radius in the l p ball gives alower bound that does not meet these numbers. Also note that lemma 2.3 is efficientto evaluate width of tori, as the filling radius of a product is the minimum of the fillingradius of each factor. wdim e B l p ( n ) We now focus on the computation of wdim e X for unit ball in finite dimensional l p .Except for a few cases, the complete description is hard to give. We start with a simpleexample. Example : Let B l ( ) be the unit ball of R for the l metric, thenwdim e B l ( ) = (cid:26) e ≥ , e < . If B l ( ) is endowed with the l p metric, then e < / p ⇒ wdim e B l ( ) = Proof.
Given any three points whose convex hull contains the origin, two of them haveto be on opposite sides, which means their distance is 2 / p in the l p metric. Hencea radial projection is possible for simplices whose vertices form sets of diameter lessthan 2 / p . Invoking lemma 2.4, FilRad ¶ B l ( ) ≥ − + / p . Lemma 2.3 concludes. Thisis specific to dimension 2 and is coherent with lemma 2.6, since, in dimension 2, l ¥ and l are isometric.An interesting lower bound can be obtained thanks to the Borsuk-Ulam theorem;as a reminder, this theorem states that a map from the n -dimensional sphere to R n hasa fibre containing two opposite points. Proposition 3.2:
Let S = ¶ B l p ( n + ) betheunitsphereofa ( n + ) -dimensionalBanachspace,then e < ⇒ wdim e S > ( n − ) / . In particular, the same statement holds for B l p ( n + ) : e < ⇒ wdim e B l p ( n + ) > ( n − ) /
2. 6 roof.
We will show that a map from S to a k -dimensional polyhedron, for k ≤ n − ,sends two antipodal points to the same value. Since radial projection is a homeomor-phism between S and the Euclidean sphere S n = ¶ B l ( n + ) that sends antipodal pointsto antipodal points, it will be sufficient to show this for S n . Let f : S n → K be an e -embedding, where K is a polyhedron, dim K = k ≤ ( n − ) / e <
2. Since anypolyhedron of dimension k can be embedded in R k + , f extends to a map from S n to R n that does not associate the same value to opposite points, because e <
2. Thiscontradicts Borsuk-Ulam theorem. The statement on the ball is a consequence of theinclusion of the sphere.Hence, wdim e B l p ( n ) always jumps from 0 to at least ⌊ n ⌋ if they are equipped withtheir proper metric. A first upper bound.
Though this first step is very encouraging, a precise evaluationof wdim can be convoluted, even for simple spaces. It seems that describing an explicitcontinuous map with small fibers remains the best way to get upper bounds. Denote by n = { , . . . , n } . Lemma 3.3:
Let B be a unit ball in a normed n -dimensional real vector space. Let { p i } ≤ i ≤ n bepointsonthesphere S = ¶ B thatarenotcontainedinaclosedhemisphere.Supposethat ∀ A ⊂ n with | A | ≤ n − ∀ l j ∈ R > ,where j ∈ n ,if k (cid:229) i ∈ A l i p i k ≤ k / ∈ A and k (cid:229) i ∈ A l i p i − l k p k k ≤ k l k p k k ≤
1. Aset p i satisfyingthisassumptiongives e ≥ Diam { p i } : = max i = j (cid:13)(cid:13) p i − p j (cid:13)(cid:13) ⇒ wdim e B ≤ n − . Proof.
This will be done by projecting the ball on the cone with vertex at the originover the n − p i . Note that n + D n be the n -simplex given by the convex hull of p , . . . , p n . We willproject the ball on the various convex hulls of 0 and n − p i . Call E the radialprojection of elements of the ball (save the origin) to the sphere, and let, for A ⊂ n , P A = { p , . . . , p n } \ { p i | i ∈ A } . In particular, P ∅ is the set of all the p i . Furthermore,denote by C X the convex hull of X . Given these notations, EC P { i } is the radial pro-jection of the ( n − ) -simplex C P { i } ( C P { i } does not contain 0 else the points wouldlie in a closed hemisphere), and EC P { i , j } are parts of the boundary of this projection.Finally, consider, again for A ⊂ n , D ′ A = C [ EC P A ∪ ] .Let s i : D ′{ i } → ∪ j = i D ′{ i , j } be the projection along p i . More precisely, we claim that s i ( p ) is the unique point of D ′{ i , j } that also belongs to L p i ( p ) = { p + l p i | l ∈ R ≥ } .Existence is a consequence of the fact that the points are not contained in an closedhemisphere, i.e. ∃ µ i ∈ R > such that (cid:229) k ∈ n µ k p k =
0. Indeed, p ∈ D ′{ i } , if p ∈ D ′{ i , j } for some j , then there is nothing to show. Suppose that ∀ j = i , p / ∈ D ′{ i , j } . Then p = (cid:229) k = i l k p k , where l k >
0. Write p i = − µ i (cid:229) k = i µ k p k . It follows that for some l , p + l p i can be written as (cid:229) k ∈ n \{ i , j } l ′ k p k with 0 ≤ l ′ k ≤ l k . Uniqueness comes from atransversality observation. D ′{ i , j } is contained in the plane generated by the set P { i , j } L p i ( p ) was to lie in that plane then the set P { j } would lie in the same plane, and P ∅ would be contained in a closed hemisphere.Thus L p i ( p ) is transversal to D ′{ i , j } . The figure below illustrates this projection in D ′{ } for n = p p p D {0,1} D {0,3} Our (candidate to be an) e -embedding s is defined by s | D ′{ i } = s i . Since on EC P { i } ∩ EC P { j } ⊂ EC P { i , j } , we see that s | D ′{ i , j } = Id and that ∪ i ∈ n D ′{ i } = B , this map is well-defined. It remains to check that the diameter of the fibres is bounded by e . We claimthat the biggest fibre is s − ( ) = ∪ i C {− p i , } , whose diameter is that of the set ofvertices of the simplex, Diam { p i } . To see this, note that for x ∈ D ′{ i , j } , the diameter of s − ( x ) attained on its extremal points (by convexity of the norm), that is x and points ofthe form x − l k p k (for k ∈ A , where A ⊃ { i , j } and x ∈ D ′ A ⊂ D ′{ i , j } ) whose norm is one.However, since x = (cid:229) l i p i for i / ∈ A and l i > k x − l k p k k = k l k p k k ≤
1, soa simple translation of s − ( x ) is actually included in s − ( ) .This allows us to have a first look at the Euclidean case. Theorem 3.4:
Let B l ( n ) be the unit ball of R n , endowed with the Euclidean metric,andlet b n ;2 : = q ( + n ) . Then,for 0 < k < n ,wdim e B l ( n ) = ≤ e , k ≤ wdim e B l ( n ) < n if b k + ≤ e < b k ;2 , wdim e B l ( n ) = n if e < b n ;2 . Proof.
First, when e ≥ Diam B l ( n ) = n = n ≥
2. Applyinglemma 2.3 to ¶ B l ( n ) ⊂ B l ( n ) yields that wdim e B l ( n ) = n if e < ¶ B l ( n ) , butFilRad B l ( n ) ≥ b n ;2 by Jung’s theorem (see [4, §2.10.41]), as any set whose diameteris less than < b n ;2 is contained in an open hemisphere ([10] shows that FilRad B l ( n ) = b n ;2 ). On the other hand, balls of dimension k < n are all included in B l ( n ) , whichmeans that wdim e B l ( k ) ≤ wdim e B l ( n ) , thanks to 2.1.d. Hence we have that wdim e B l ( n ) ≥ k whenever b k + ≤ e < b k ;2 . This proves the lower bounds.8he vertices of the standard simplex satisfy the assumption of lemma 3.3: thanks tothe invariance of the norm under rotation we can assume p = ( , , . . . , ) . The other p i will all have a negative first coordinate, and so will any positive linear combination.Substracting l p will be norm increasing. As the diameter of this set is b n ;2 , lemma3.3 gives the desired upper bound.Let us now give an additional upper bound for the 3-dimensional case: Proposition 3.5:
If 1 ≤ p < ¥ ,then e ≥ ( ) / p ⇒ wdim e B l p ( ) ≤ Proof. In R there is a particularly good set of points to define our projections. Theseare p = - p ( , , ) , p = - p ( , -1 , -1 ) , p = - p ( -1 , , -1 ) and p = - p ( -1 , -1 , ) . Let x = l p , where l ∈ [ , ] , and suppose k l p − l p k l p ≤ l ∈ R ≥ . We have tocheck that l ≤
1. Suppose l >
1, then 1 ≥ k l p − l p k l p = ( l + l ) p + ( l − l ) p = l p [ ( + t ) p + ( − t ) p ] , where t = l / l . The function of t has minimal value1, which gives l ≤ x = l p + l p is of norm less than 1, where without loss ofgenerality we assume l ≥ l , and k l p + l p − l p k l p ≤ k x k l p ≤ ≥ ( l + l ) p + ( l − l ) p so ( l − l ) p ≤ − ( l + l ) p + ( l − l ) p ≤
1. If l >
1, then1 ≥ k l p + l p − l p k l p = ( l + l + l ) p + ( l − ( l − l )) p + ( l + ( l − l )) p . However, l p ≤ ( l + l + l ) p + l p ≤ ( l + l + l ) p + ( l − ( l − l )) p + ( l + ( l − l )) p ≤ . Using that f ( t ) = ( + t ) p + ( − t ) p has minimum 2 for t ∈ [ , ] . These argumentscan be repeated for any indices to show that the points p i , where i = , , ( p i ) = ( ) / p For certain dimensions, a set of points that allows to build projections with smallfibers can be found. Their descriptions require the concept of Hadamard matrices ofrank N ; these are N × N matrices, that will be denoted H N , whose entries are ± H N · H tN = N Id. It has been shown that they can only exist when N = | N , and it is conjectured that this is precisely when they exist. Up to a permutationand a sign, it is possible to write a matrix H N so that its first column and its first rowconsist only of 1s. It is quite easy to see that two rows or columns of such a matrixhave exactly N / Definition 3.6:
Let H N be a Hadamard matrix of rank N , and let, for 0 ≤ i ≤ N , h i be the i th row of the matrix without its first entry (which is a 1). Then the h i form aHadamard set in dimension N −
1. 9hese N elements, normalised so that k h i k l p ( N − ) =
1. When so normalised, theirdiameter (for the l p metric) is 2 − / p ( + N − ) p . Since (cid:229) h i =
0, by orthogonality ofthe columns with the column of 1 that was removed, we see that they are not containedin an open hemisphere. The set of points in the preceding proposition was given by aHadamard matrix of rank 4, and when p = Proposition 3.7:
SupposethereexistsaHadamardmatrixofrank n + e ≥ + n ⇒ wdim e B l ( n ) ≤ n − . Proof.
Let the h i be as above, and N = n +
1. Note that for i = j , h i and h j have N opposed coordinates, and N − l i h i − l j h j has always a big-ger l norm than any of its two summands. Indeed, the coefficients c j of the vector (cid:229) i ∈ A l i h i where the contribution of h k reduces (cid:12)(cid:12) c j (cid:12)(cid:12) are in lesser number than those thatget increased. Since the l norm is linear, the magnitude of the c j getting smaller is notrelevant, only their number.We conclude by applying lemma 3.3, as Diam l ( h i ) = + N − .Note that in dimension higher than 3 and for p >
2, Hadamard sets no longer satisfythe assumption of lemma 3.3.
Further upper bounds for wdim e B l p ( n ) . The projection argument still works fornon-Euclidean spheres. It can also be repeated, though unefficiently, to construct mapsto lower dimensional polyhedra.
Proposition 3.8:
For 1 < p < ¥ , consider the sphere B l p ( n ) with its natural metric.Then,for n − < k < n , ∃ c k , n ; p ∈ [ , ) suchthat c k , n ; p ≥ c k + , n ; p ,andwdim e B l p ( n ) ≤ k if e ≥ c k , n ; p . Furthermore c n − , n ;2 = b n ;2 Proof.
This proposition is also obtained by constructing explicitly maps that reducedimension (up to n − j for j < n + ) and whose fibres are small. Unfortunately, nothingindicates this is optimal, and the size of the preimages is hard to determine. We willabbreviate B : = B l p ( n ) .We proceed by induction, and keep the notations introduced in the proof of lemma3.3. The p i that are used here are the vertices of the simplex; they need to be renor-malised to be of l p -norm 1, but note that multiplying them by a constant has actuallyno effect in this argument. Also note that the sets D ′ A are not the same for different p , since they are constructed by radial projection to different spheres. The keys tothis construction are the maps s j ; { i ,..., i j } : D ′{ i ,..., i j } → ∪ m / ∈{ i ,..., i j } D ′{ i ,..., i j , m } given byprojection along the vectors j (cid:229) l = p i l . Call s the function s from lemma 3.3, then, for10 > s j : B → ∪ { i ,..., i j + }⊂ n D ′{ i ,..., i j + } is obtained by composing, on appropriate do-mains, s j ; { i ,..., i j } with s j − . Since s j ; i ,..., i j are equal to the identity when their domainintersect, and their union covers the image of s j − , the map is again well-defined. Itremains only to calculate the diameter of the fibres. At 0 the fibre is s − j ( ) = ∪ { i ,..., i j }⊂ n {− ( l + . . . + l j ) p i − ( l + . . . + l j − ) p i − . . . − l p i j | l i ∈ R ≥ } . Whereas for a given x ∈ D ′ A in the image (that is A contains at least j elements), x canalso be written down as a combination (cid:229) l i p i , for i / ∈ A and l i ∈ R > . We have s − j ( x ) = ∪ { i ,..., i j }⊂ A { x − ( l + . . . + l j ) p i − ( l + . . . + l j − ) p i − . . . − l p i j | l i ∈ R ≥ } . If we set c k , n = sup x ∈ s j ( B ) Diam s − n − k ( x ) , then when e ≥ c k , n , wdim e B l ( n ) ≤ k . It is possibleto determine two simple facts about these numbers. First, they are non-increasing c k , n ≥ c k + , n , which is obvious as the construction is done by induction, the size of thefiber of maps to lower dimension is bigger than for maps to higher dimension.Second, they are meaningful: c k , n <
2. Indeed, when p = , ¥ , c k , n = s − n − k ( x ) contains opposite points, which is a linear condition. When x =
0, by convexityof the distance, the points on which the diameter can be attained are at the boundary of s − j ( x ) . Say Y is the set of those point except x . The distance from Y to x is at mostone, while the diameter of Y is bounded. Indeed, there is a cap of diameter less than 2that contains all the p i but one. The biggest diameter of such caps is also less than 2and bounds Diam Y .Any point of the fibre at 0 is a linear combination of the vertices p i , and there is onlyone linear relation between these, namely (cid:229) p i =
0. As long as j < n + ( i.e. k > n − )there are not enough p i in any two sets that form s − j ( ) to combine into the requiredrelations, but as soon as j exceeds this bound, opposite points are easily found.For B l p ( n ) , where 1 < p < ¥ , we used the regular simplex to describe our projec-tions, though nothing indicates that this choice is the most appropriate. In fact, manysets of n + x to be too large, as in the assumption of lemma 3.3). Furthermore,there is in general no reason for c n − , n ; p to coincide with a lower bound, or even to bedifferent from other c k ; p , thus we cannot always insure that n − ∈ wspec ( B l p ( n ) , l p ) . The lowest non zero element of wspec . Before we return to the general l p case, no-tice that together proposition 3.2 and theorem 3.4 give a good picture of the functionwdim e B l ( n ) . It equals n for e < b n ;2 = c n − , n ;2 , then n − b n ;2 ≤ e < b n − . After-wards, I could not show a strict inequality for the c k , n ;2 , but even if they are all equal,wdim e B l ( n ) takes at least one value in ( n − , n + ) ∩ Z . Then ,when e ≥
2, it dropsto 0. 11or odd dimensional balls, there is a gap between the value given by proposition3.2 and the lowest dimension obtained by the projections introduced above. Say B is ofdimension 2 l + e less than but sufficiently close to 2, then on one hand we knowthat wdim e B ≥ l , while on the other wdim e B ≤ l +
1. It is thus worthy to ask whetherone of these two methods can be improved, perhaps by using extra homological in-formation on the simplices in the proof of proposition 3.2 ( e.g. if its highest degreecohomology is trivial then a k -dimensional polyhedron is embeddable in R k , see [5]). Remark : Such an improvement is actually available when n =
3: if the 2-dimensionalsphere maps to a 1-dimensional polyhedron ( i.e. a graph), the map lifts to the universalcover, a tree K . Hence K is embeddable in R , and, for 1 < p < ¥ . e < ⇒ wdim e B l p ( ) ≥ , can also yield lower boundsfor the diameter of fibres for maps to graphs ( i.e. Lower bounds for wdim e B l p ( n ) . The remainder of this section is devoted to the im-provement of lower bounds, using an evaluation of the filling radius as a product oflemma 2.4, and a short discussion of their sharpness.We shall try to find a lower bound on the diameter of n + l p unitsphere that are not in an open hemisphere; recall that points f i are not in an openhemisphere if ∃ l i such that (cid:229) l i f i =
0. A direct use of Jung’s constant (defined as thesupremum over all convex M of the radius of the smallest ball that contains M dividedby M ’s diameter) that is cleverly estimated for l p spaces in [9] does not yield the resultlike it did in the Euclidean case. This is due to the fact that there are sets of n + ( , . . . , ) , (cid:18) − n − , . . . , − n − , (cid:19) , . . . , and (cid:18) , − n − , . . . , − n − (cid:19) is such an example for l ¥ , and deforming it a little can make it work for the l p case, p finite but close to ¥ . However, a very minor adaptation of the methods given in [9] issufficient.First, we introduce norms for the spaces of sequences (and matrices) taking val-ues in a Banach space E . Let a i ∈ R ≥ be such that n (cid:229) i = a i = a thissequence of n + E p , a be the space of sequences made of n + E and consider the l p norm weighted by a : k x k E p , a = (cid:0) (cid:229) i a i k x i k pE (cid:1) / p where x = ( x , . . . , x n ) . On the other hand, E p , a shall represent the space of matrices whoseentries are in E , with the norm (cid:13)(cid:13) ( x i , j ) (cid:13)(cid:13) E p , a = (cid:0) (cid:229) i , j a i a j (cid:13)(cid:13) x i , j (cid:13)(cid:13) pE (cid:1) / p . Now define, for E , E ′ Banach spaces based on the same vector space and for 1 ≤ s , t ≤ ¥ , the linearoperator T : E s , a → E ′ t , a by ( x i ) ( x i − x j ) .12 heorem 3.11: Consider a vectorspace on which two normsaredefined, and denoteby E , E theBanachspacetheyform. Let f i ∈ E ∗ , 0 ≤ i ≤ n ,besuchthat k f i k E ∗ = l i ∈ R ≥ suchthat (cid:229) l i f i = (cid:229) l i =
1. Let Diam E ∗ ( f ) = sup ≤ i , j ≤ n (cid:13)(cid:13) f i − f j (cid:13)(cid:13) E ∗ be the diameter ofthissetwithrespecttotheothernorm. Then,for a i = l i ,Diam E ∗ ( f ) ≥ ≤ s , t ≤ ¥ (cid:18) + n (cid:19) / t ′ sup E k T k − ( E ) s , a → ( E ) t , a . Proof.
As the f i are not in an open hemisphere, real numbers l i ∈ R ≥ such that (cid:229) l i = (cid:229) l i f i = k f i k E ∗ =
1, there also exist x i ∈ E such that f i ( x i ) = k x i k E =
1. The remark on which the estimation relies is, as in [9],2 = n (cid:229) i , j = l i l j ( f i − f j )( x i − x j ) . Choosing a i = l i , this equality can be rewritten in the form 2 = ( T f )( T x ) , where T x ∈ ( E ) t , a and T f ∈ (( E ) t , a ) ∗ = ( E ∗ ) t ′ , a , and thus 2 ≤ k T f k ( E ∗ ) t ′ , a k T x k ( E ) t , a .Notice that (cid:229) i = j a i a j = n (cid:229) i = a i ( − a i ) ≤ − n + = nn + , because k a i k l ( n + ) = ⇒ k a i k l ( n + ) ≥ ( n + ) − / . We can isolate the required di-ameter: k T f k ( E ∗ ) t ′ , a = (cid:0) n (cid:229) i = a i a j (cid:13)(cid:13) f i − f j (cid:13)(cid:13) t ′ E ∗ (cid:1) / t ′ ≤ Diam E ∗ ( f ) (cid:0) (cid:229) i = j a i a j (cid:1) / t ′ ≤ Diam E ∗ ( f ) (cid:0) nn + (cid:1) / t ′ . On the other hand, k x i k E =
1, consequently k x k ( E ) s , a =
1, so we bound k T x k ( E ) t , a ≤ k T k ( E ) s , a → ( E ) t , a . The conclusion is found by substitution of the estimates for the norms of
T f and
T x .We only quote the next result, as there is no alteration needed in that part of theargument of Pichugov and Ivanov.
Theorem 3.12: (cf. [9,thm2])if 1 ≤ p ≤ , k T k ( l p ( n )) ¥ , a → ( l p ( n )) p , a ≤ / p (cid:0) nn + (cid:1) / p − / p ′ , if 2 ≤ p ≤ ¥ , k T k ( l p ( n )) ¥ , a → ( l p ( n )) p , a ≤ / p ′ . A simple substitution in theorem 3.11, with E = E = l p ( n ) , s = ¥ and t = p ,yields the desired inequalities. 13 orollary 3.13: Let f i , 0 ≤ i ≤ n , be points on the unit sphere of l p ( n ) that are notincludedinanopenhemisphere,thenif 1 ≤ p ≤ , Diam l p ( n ) ( f ) ≥ / p ′ (cid:0) + n (cid:1) / p , ( ∗ ) if 2 ≤ p ≤ ¥ , Diam l p ( n ) ( f ) ≥ / p (cid:0) + n (cid:1) / p ′ . ( ∗∗ ) Remark : Before we turn to the consequences of this result on wdim e , note thatthere are examples for which the first inequality is attained. These are the Hadamardsets defined in 3.6. When normalised to 1, they are not included in an open hemisphereand of the proper diameter. Hence, when a Hadamard matrix of rank n + ( ∗ ) is optimal. Nothing so conclusive can be said for other dimensions, see theargument in example 3.1. I ignore if there are cases for which ( ∗∗ ) is optimal, though itis very easy to construct a family F n ∈ ( B l p ( n ) ) n + such that Diam F n → / p as n → ¥ .In particular for p = ¥ , the points given in (3.10) but by substituting − n − instead of theentries with value − n − , is a set that is not contained in an open hemisphere and whosediameter is nn − , which is close to the bound given. Somehow, this case, is also the onewhere the use of lemma 2.4 results in a bound that is quite far from the right value ofwdim, cf. lemma 2.6. This might not be so surprising as sets with small diameter on l p balls seem, when p >
2, to differ from sets satisfying the assumption of lemma 3.3.Still, by lemma 2.4 we obtain the following lower bounds on wdim:
Corollary 3.15:
Let b k ; p bedefinedby b k ; p = / p ′ (cid:0) + k (cid:1) / p when1 ≤ p ≤ b k ; p = / p (cid:0) + k (cid:1) / p ′ if 2 ≤ p < ¥ . Then,for 0 < k ≤ n , e < b k ; p ⇒ wdim e B l p ( n ) ≥ k . Proof.
Let Y = ¶ B l p ( n ) . Since the convex hull of a set of n + Y will not contain the origin if the diameter of the set is larger than b n ; p , lemma 2.4ensures that FilRad Y ≥ b n ; p /
2. We then use lemma 2.3 for Y to conclude.These inequations might not be optimal, proposition 3.2 for example is alwaysstronger when k < ⌊ n ⌋ .In dimension n , B l ¥ ( n ) n − / p ⊂ B l p ( n ) yields that e < n − / p ⇒ wdim e B l p ( n ) = n whichimproves corollary 3.15 as long as p ≥ ln ( n n + ) ln ( nn + ) . However, when p =
1, and H n + is a Hadamard matrix, these estimates are as sharp aswe can hope since the lower bound meets the upper bounds. Corollary 3.16:
SupposethereisaHadamardmatrixofrank n +
1. Then,for 0 ≤ k < n , wdim e B l ( n ) = ≤ e , max ( n − , k ) ≤ wdim e B l ( n ) < n if ( + k + ) ≤ e < ( + k ) , wdim e B l ( n ) = n if e < ( + n ) . ≤ p ≤
2. In particular, thanks to remark 3.9, this gives acomplete description of the 3-dimensional case for such p . Corollary 3.17:
Let p ∈ [ , ] ,thenwdim e B l p ( ) = ≤ e , ( ) / p ≤ e < , e < ( ) / p . When p >
2, all that can be said is that the value of e for which wdim e B l p ( ) dropsfrom 3 to 2 is in the interval [ ( ) − / p , ( ) / p ] .This last corollary is special to the 3-dimensional case, which happens to be adimension where there exist a Hadamard set, and where the Borsuk-Ulam argumentcan be improved to rule out maps to n − -dimensional polyhedra. For example, in the2-dimensional case, a precise description is not so easy. Indeed, thanks to example 3.1and using the inclusion of B l ( ) ⊂ B l p ( ) , we know that wdim e B l p ( ) = e ≥ / p .On the other hand, the inclusion of B l ¥ ( ) − / p ⊂ B l p ( ) gives e ≥ / p ′ ⇒ wdim e B l p ( ) = e ≥ max ( / p , / p ′ ) ⇒ wdim e B l p ( ) = . These simple estimates in dimension 2 are better than corollary 3.15 as long as p ≤ − ln3ln2 or p ≥ ln ( ) / ln ( ) . I doubt that any of these estimations actually gives thevalue of e where wdim e B l p ( ) drops from 2 to 1.All the results of this section can be summarised to give theorem 1.4. Here aretwo depictions of the situation. Gray areas correspond to possible values, full lines toknown values and dotted line to bounds. (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) b {n;p}n2 wdm e e cn b {n;p}n2 wdm e e cn The left-hand plot is for euclidean case, or the case where p = n − c ⌈ n / ⌉ , n ; p is abbreviated by c .The case of dimension 3 is described in corollary 3.17.It is not expected that n − be in wspec when n is odd, nor is it expected that thelower bounds b k ; p be sharp for B l p ( n ) when k < n . References [1] Michel Coornaert.
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