A uniform lower bound on the norms of hyperplane projections of spherical polytopes
aa r X i v : . [ m a t h . F A ] S e p A UNIFORM LOWER BOUND ON THE NORMS OF HYPERPLANEPROJECTIONS OF SPHERICAL POLYTOPES
TOMASZ KOBOS
Abstract.
Let K be a centrally symmetric spherical polytope, whose vertices form a n − net inthe unit sphere in R n . We prove a uniform lower bound on the norms of hyperplane projections P : X → X , where X is the n -dimensional normed space with the unit ball K . The estimate isgiven in terms of the determinant function of vertices and faces of K . In particular, if N ≥ n n and K = conv {± x , ± x , . . . , ± x N } , where x , x , . . . , x N are independent random points distributeduniformly in the unit sphere, then every hyperplane projection P : X → X satisfies the inequality || P || ≥ c n N − n − (for some explicit constant c n ), with the probability at least − N . Introduction
Let X be a real normed space of dimension at least . A linear and continuous mapping P : X → X is called a projection , if it satisfies the equation P = P . By a hyperplane projection weshall mean a projection with the image of codimension .Projection is a very old concept in mathematics and a basic notion of the approximation theory,as it provides an approximation of the identity operator on a subspace, by a linear operator definedon the whole space. For this reason, one often seeks for a projection with the smallest possibleoperator norm, as the smaller norm yields a better approximation. Such a projection P is called a norm-minimal projection from X onto the image of P .Norm-minimal projection were studied by a lot of different authors in a great variety of contexts(see for example [1], [8], [9] [10], [14], [17], [20], [21], [22]). The so-called projection constants were studied most extensively. Projection constants play a profound role in the functional analysisand the local theory of Banach spaces, as they are deeply connected with some other importantnumerical invariants of Banach spaces. We refer to Chapter 8 in a monograph [26] for a broaderpicture on the theory of the projection constants. In terms of the norm, the best possible situationhappens, when there exists a projection of norm onto a given subspace. In this case, we say thata subspace is -complemented in the given space.By the Hahn-Banach theorem, every -dimensional subspace is -complemented. For this reason,we shall call a projection non-trivial , if its image has dimension at least , and it is different fromthe whole space. In a Hilbert space, every subspace is -complemented by means of the orthogonalprojection. Conversely, it is well-known, that if every subspace of a given Banach space X , satisfying dim X ≥ , is -complemented, then X is isometric to a Hilbert space (see for example [19]). Still,most of the classical spaces posses some -complemented subspace of dimension at least , even ifthey are not necessarily Hilbert. Study of -complemented subspaces of Banach spaces has a longhistory and there is a large volume of published research on this topic.Bosznay and Garay proved in [4] that, in the context of a normed spaces of given dimension n ≥ , this is, in fact, a very rare instance, to posses some non-trivial projection with the norm .It turns out, that the set of n -dimensional normed spaces, for which every non-trivial projection P : X → X has norm strictly larger than , is open and dense in the set of all n -dimensional normed Mathematics Subject Classification.
Primary 47A58, 41A65, 47A30, 52A21, 52A22.
Key words and phrases.
Minimal projection, Finite-dimensional normed space, Spherical polytope, Randompolytope. paces. This somewhat reminds of the well-known fact, that the set of continuous and nowheredifferentiable functions forms an open and dense subset of the set of continuous functions. Moreover,this naturally raises a question of establishing some explicit, uniform lower bound on the norms ofprojections of a given space, which is strictly greater than . Thinking more globally, it is naturalto define the constant ρ n as the largest positive number, for which there exists an n -dimensionalnormed space X , such that every non-trivial projection P : X → X satisfies || P || ≥ ρ n . Thefact, that ρ n is positive, follows immediately from the result of Bosznay and Garay and a standardcompactness argument. By a result of [8], on every two-dimensional subspace there is always aprojection with the norm at most , so obviously we have ρ n ≤ for every n ≥ .The question of estimating ρ n was stated in [4] as Problem 2. To this time, it seems that nopositive lower bounds on ρ n are known. This may be related to the fact, that lower bounds for thenorms of projections were studied mostly in the case of specific subspaces, rather than uniformly.Nevertheless, some remarkable results related to the uniform lower bounds were obtained. Gluskinin [13] and Szarek in [25] used norms generated by random polytopes to establish such lower bounds,but only for projections with the rank in a specific range. Later, a similar construction was providedalso in [23]. All of these results give estimates of the following type: there exists an n -dimensionalnormed space X , such that for every projection P : X → X with the rank m in an interval of theform [ αn, βn ] (where < α < β < are constants), we have || P || ≥ C √ m (for some constant C depending on α and β ). Asymptotically speaking, this is best possible up to a constant, as thefamous result of Kadec and Snobar (see [17]) yields the inequality || P || < √ m .A deeply profound role, that random polytopes play in the modern high-dimensional geometry,has been started with a pioneering previous work of Gluskin in [12], who used them to prove thatthe asymptotic order of the diameter of the Banach-Mazur compactum is linear. After that, manydifferent important applications of the random objects in the high-dimensional geometry have beenestablished, including the examples above.It does not seem possible to apply those methods directly to projections with the rank not inthe interval of the form [ αn, βn ] . In this case, the examples are generally lacking. However, someresults were obtained in [18] for the case of hyperplane projections. For each n ≥ let us define aconstant ρ Hn as the largest positive number, for which there exists an n -dimensional normed space X , such that every hyperplane projection P : X → X satisfies || P || ≥ ρ Hn . Obviously, we have ρ Hn ≥ ρ n for every n ≥ . Moreover, by a result of Bohnenblust (see [2]), every hyperplane admits aprojection of norm at most − n , and therefore ρ Hn ≤ − n for every n ≥ . In [18] a uniform lowerbound on the norms of the hyperplane projections was provided in the case of the space X being arather general subspace of the ℓ m p space (where p ≥ is an integer). In consequence, we have ρ Hn ≥ (cid:16) n + 3) (cid:17) − n +3) , (1)for every n ≥ . This implies an asymptotic lower bound on ρ Hn of the form ρ Hn ≥ exp( − Cn log n ) , (2)for some absolute constant C > .The aim of this paper, is to study uniform lower bounds for the norms of hyperplane projectionsin the setting of the spherical polytopes. Our main result gives such an explicit, uniform lowerbound for a broad class of normed spaces, with the unit ball being a symmetrical spherical polytopewhose vertices form a n − net in the unit sphere. By || · || we shall always mean the Euclidean normin R n . By S n = { x ∈ R n : || x || = 1 } we denote the Euclidean unit sphere in R n . If K ⊂ R n isa convex polytope, then by a term face we shall mean only ( n − -dimensional face (facet) of K .Two faces of K are called non-neighbouring if their intersection is empty. For a given ε > , a set X ⊂ S n is called an ε -net if for every point p ∈ S n , there exists x ∈ X such that || x − p || ≤ ε .Throughout the paper, we assume that n ≥ is a positive integer. Our main result goes as follows. heorem 1.1 (General lower bound for the spherical polytopes) . Let N be a positive integer and α, β positive real numbers. Suppose that points x , x , . . . , x N ∈ S n satisfy the following conditions:(1) Vertices of a convex polytope K = conv {± x , ± x , . . . , ± x N } form a n -net in S n .(2) For any ≤ i < i < . . . < i n ≤ N we have | det( x i , x i , . . . , x i n ) | ≥ α .(3) For any n pairwise non-neighbouring faces of K , given by the vectors f , f , . . . , f n ∈ S n , wehave | det( f , f , . . . , f n ) | ≥ β .Let X be the n -dimensional normed space with the unit ball K . Then, every hyperplane projection P : X → X satisfies || P || ≥ C n α β, where C n = 2 n − · n n − √ n − . It is easy to see, that for a generic symmetric polytope with vertices in S n , the determinantfunction does not vanish on any subset of vertices or pairwise non-neighbouring faces. Thus, ageneric symmetric polytope, with vertices forming a n -net in the unit sphere, has all hyperplaneprojections with the norm greater than . This is clearly expected by the result of Bosznay andGaray. It also leads to the question of estimating the measure of the spherical polytopes satisfyingsome explicit, uniform lower bound on the norms of hyperplane projection. Points x , x , . . . , x N ∈ S n will be called random points , if they are distributed independently and uniformly in S n . Inthe next result, we provide a uniform lower bound for the norms of hyperplane projections, whichholds with a large probability for a random spherical polytope. The result states, that for thesymmetric convex hull of N ≥ n n random points in S n , the corresponding normed space has allhyperplane projections with norm greater than c n N − n − (for some specific constant c n ), withthe probability at least − N . Theorem 1.2 (Lower bound for random spherical polytopes) . Let N ≥ n n be a positive integer.Let x , x , . . . , x N ∈ S n be random points and let X = ( R n , || · || X ) be the n -dimensional normedspace with the unit ball B X = conv {± x , ± x , . . . , ± x N } . Then, the probability that every hyperplaneprojection P : X → X satisfies || P || ≥ C n N (cid:18) Nn (cid:19) (cid:18) n N n (cid:19) ! − , where C n = n · √ n − · n · n n − · ( n − · e n , is at least − N . We can say that the result above quantifies the original result of Bosznay and Garay (in thehyperplane setting) in two different ways. It gives a uniform lower bound on the norms of projections,but it also estimates the measure of the spherical polytopes with given number of vertices, whichsatisfy it. In the three-dimensional case, the estimate on the norm can be strengthened to cN − ,with some explicit constant c > – see Remark 4.1. Actually, for an arbitrary fixed dimension n ≥ and any fixed ε > , the lower bound of c n N − n − can be improved to c n,ε N − n − − ε . SeeRemark 4.3 for further details. We also note, that even if we work with random polytopes, ourmethods are different than those from previously mentioned papers [13], [23], [25]. This may stemfrom the fact that the hyperplane case seems to be rather dissimilar to the case of projections withthe rank depending linearly on n . In particular, we do not rely on some more advanced variants ofthe concentration of measure, but we use only basic probabilistic tools, such as Markov’s inequality.However, we do rely on some more involved results concernerning the random polytopes and some f them were developed recently. Let us also remark, that we took some care, to keep all constantsappearing in our estimates as explicit as possible.Since we consider polytopes approximating the unit sphere very well, the corresponding normis close to the Euclidean norm and we are working rather locally around it. Therefore, as theorthogonal projection has norm , it is not reasonable to expect that our results will yield anoptimal lower bound on the constant ρ Hn . By taking N = n n in Theorem 1.2 we get a lower bound ρ Hn ≥ exp( − Cn ) , which is worse than the lower bound (2) obtained in [18]. However, if one is interested only inasymptotics of ρ Hn , it is not difficult to recover the lower bound (2), with a minor modification ofthe proof of Theorem 1.2. This may be quite surprising, that two completely different approachesyield exactly the same asymptotic bound on p Hn . See Remark 4.4 for further details. We also notethat the estimate (1) from [18] works only for n ≥ . Thus, our result gives a first non-trivial boundfor the three-dimensional constant ρ H = ρ . See Remark 4.2 for some numerical estimate on that ρ can be deduced with our approach.The paper is organized as follows. In Section 2 we prove Theorem 1.1. In Section 3 we establishTheorem 1.2 by applying Theorem 1.1 in combination with several auxiliary results concerningrandom polytopes. The paper is concluded in Section 4, where some further remarks related to ourresults, are provided.2. Proof of the general lower bound for the spherical polytopes
In this Section we prove Theorem 1.1. We start with some simple auxiliary results.
Lemma 2.1.
Let N be a positive integer and suppose that a set { x , x , . . . , x N } ⊂ S n is an ε -netfor some < ε < . Then, every face of a convex polytope K = conv { x , x , . . . , x N } has diameternot greater than ε and an inclusion (1 − ε ) S n ⊂ K holds.Proof . Let F be any face of K and let x be the center of the spherical cap determined by F .Clearly, x is equidistant to the vertices of F . Let us denote this distance by d . We have d ≤ ε . Bythe triangle inequality, the distance between any two vertices of F is at most d ≤ ε , which showsthe first claim. For the second claim, let h be the distance between x and the hyperplane contaning F . Clearly h ≤ d ≤ ε . Moreover, the hyperplane tangent in x to S is parallel to the hyperplanecontaining F . Since − h ≤ − ε , it is clear that S ⊂ − ε K and the proof is complete. (cid:3) Lemma 2.2.
Let x , x , . . . , x n ∈ S n be such that | det( x , x , . . . , x n ) | ≥ α for some α > . Then,for any v ∈ S n we have max ≤ i ≤ n |h x i , v i| ≥ αn . Proof.
Assume on the contrary, that for for some v ∈ S n and for each ≤ i ≤ n we have that h x i , v i = r i ∈ ( − αn , αn ) . Let r = ( r , r , . . . , r n ) ∈ R n . Then || r || < q nα n = α √ n . Hence, from theCramer’s rule and the Hadamard inequality it follows that | v | = (cid:12)(cid:12)(cid:12)(cid:12) det( r, x , . . . , x n )det( x , x , . . . , x n ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ || r || α < √ n . In the same way we prove that | v i | < √ n for ≤ i ≤ n . Hence v + v + . . . + v n < n · n = 1 , which gives the desired contradiction. (cid:3) Now we are ready to prove Theorem 1.1.
Proof of Theorem 1.1.
We denote by || · || X the norm of X and by || · || X ∗ the dual norm. By Lemma 2.1 every face of K is of a diameter not greater than d = n and S n ⊂ n n − K . Moreover, by the second assumption, he convex polytope K is simplicial , which means that its every face is an ( n − -dimensionalsimplex. Let Y ⊂ X be an arbitrary ( n − -dimensional subspace. Suppose that P : X → X is aprojection with the image Y and norm less than C n α β . Let w ∈ ker P with || w || = 1 and let Y = { x ∈ R n : h x, v i = 0 } with || v || = 1 . Assume that Y has non-empty intersection with the faces F , F , . . . , F k , − F , − F , . . . , − F k of K , given by the vectors f , f , . . . , f k , − f , − f , . . . , − f k ∈ S n .Let us call a face F i a bad face, if there does not exist a vector z ∈ Y ∩ F i such that dist( z, bd F i ) ≥ s (the boundary bd F i is considered in the hyperplane containing F i ), where s = 2 n − α n · √ n − · d n − . Otherwise, a face F i is called a good face. We shall prove the following claim. Claim 1 . If F is a bad face, then there exists a vertex a of F such that |h a, v i| < αn . Indeed, let F = conv { a , a , . . . , a n } be a bad face. The region { x ∈ F : dist( x, bd F ) ≥ s } is a simplex F ′ , positively homothetic to F (soon, we shall see that F ′ is non-empty). As thehyperplane Y does not intersect F ′ , the simplex F ′ lies in one of the open half-spaces determinedby Y . Without loss of generality, let us assume that a vertex a of F lies in the opposite (closed)half-space – as Y has a non-empty intersection with F , there are vertices in both closed half-spaces.Let f be the face of F not containing a (thus f has dimension n − . Then, it is clear that Y intersects the parallelotope P = { x ∈ F : dist( x, f ) ≤ s for every face f = f of F } . Let a ′ be a vertex of F ′ corresponding to a . Then a ′ ∈ P and || a − x || ≤ || a − a ′ || for some x ∈ P ∩ Y . We shall now prove an upper estimate on the distance || a − a ′ || . Let < k < be thehomothety ratio of F and F ′ and let r be the inradius of F . The homothety center c is the incenterof both F and F ′ . In particular, kr + s = r , which gives us an equality k = r − sr . Furthermore, || a − c || = || a − a ′ || + || a ′ − c || = || a − a ′ || + k || a − c || , which yields || a − a ′ || = (1 − k ) || a − c || = sr || a − c || ≤ dsr . (3)We shall now establish a lower bound on the inradius r . Let us observe that vol conv { , a , . . . , a n } = | det( a , . . . , a n ) | n ! ≥ αn ! , but on the other hand, vol conv { , a , . . . , a n } = hn · rn − S + S + . . . + S n ) , where h denotes the distance of the hyperplane determined by F to the origin, and S i for ≤ i ≤ n denote the ( n − -dimensional volumes of the faces of F . Each face of F is an ( n − -dimensionalsimplex with the edge length not greater than d . Thus, it is well-known, that under these constraints,for each ≤ i ≤ n , volume S i is not greater than that of an ( n − -dimensional regular simplex ofedge length d , which is equal to √ n − n − · n − · d n − . Combining the two previous estimates with an obvious inequality h < , we get a lower bound r > ( n − αn ! · ( n − · n − √ n − · d n − = 2 n − αn · √ n − · d n − . et us note here, that this shows in particular, that the region F ′ is non-empty, as by the aboveinequality we clearly have s < r . Now, coming back to (3) we obtain || a − a ′ || ≤ dsr < ds · n · √ n − · d n − n − α = αn . Hence |h a , v i| = |h a − x, v i| ≤ || a − x || ≤ || a − a ′ || < αn , which proves Claim 1.Now, we shall establish a similar claim for the good faces. Claim 2. If F is a good face, given by the vector f ∈ S n , then |h w, f i| < βn .Since F is a good face, there exists a vector z ∈ Y ∩ F , such that dist( z, bd F ) ≥ s . Thismeans, that a ball B ( z, s ) with the center z and radius s , intersected with the boundary of K , isan ( n − -dimensional Euclidean ball contained in F . Let us take any real number λ satisfying | λ | ≤ s . Then, we have |||| z + λw || X z − ( z + λw ) || X = || ( || z + λw || X − z − λw || X ≤ ||| z + λw || X − | || z || X + | λ ||| w || X = ||| z + λw || X − || z || X | + | λ ||| w || X ≤ | λ ||| w || X + | λ ||| w || X = 2 | λ ||| w || X ≤ | λ | − n ≤ | λ | . Moreover, || z + λw || X ≥ − | λ ||| w || X ≥ − − n | λ | ≥ − | λ | . By combining two previous estimates we obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z − z + λw || z + λw || X (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ≤ | λ | − | λ | ≤ s, where the last inequality follows easily from the inequality | λ | ≤ s . Finally, we have that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z − z + λw || z + λw || X (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z − z + λw || z + λw || X (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ≤ s. This shows that for | λ | ≤ s , the vector z + λw || z + λw || X belongs to the intersection of a ball B ( z, s ) withthe boundary of K and therefore to the face F . In consequence, we have || z + λw || X = h z + λw, ˜ f i , where ˜ f = f || f || X ∗ . Thus λ h w, ˜ f i = h z + λw, ˜ f i = || z + λw || X > || P ( z + λw ) || X C n α β = || z || X C n α β = 11 + C n α β . Hence, λ h w, ˜ f i ≥ − C n α β C n α β ≥ − C n α β By taking λ = ± s and using the fact that || ˜ f || ≥ || f || , we get |h w, f i| ≤ |h w, ˜ f i| ≤ C n α βs = βn , by the definitions of s and C n . This proves Claim 2.Now, with both Claims in our disposal, we can finish the proof of Theorem 1.1. We take anypoint x ∈ Y with || x || X = 1 and any two-dimensional subspace V ⊂ Y such that x ∈ V . Let usconsider a two-dimensional curve (a broken path), lying in V ∩ bd K , which connects x and − x . learly, its Euclidean length is greater than || x || ≥ n − n || x || X = n − n . This means, that we canfind points p , p , . . . , p n − on this curve such that, for i = j we have || p i − p j || ≥ n − n (2 n − > n = 2 d. Every point p i lies in the boundary of K , and thus in some face of K . Let F i be any face of K suchthat p i ∈ F i . Note for i = j , faces F i and F j are non-neighbouring. Indeed, if u ∈ F i ∩ F j , then d = 1 n < || p i − p j || ≤ || p i − u || + || u − p j || ≤ d, which is a contradiction. It is clear, that in the set { F , F , . . . , F n − } there are at least n badfaces or at least n good faces. If there are n good faces, then by Claim 1, we get existence of n vertices a , a , . . . , a n of K , such that |h a i , v i| < αn for any ≤ i ≤ n . This is an immediatecontradiction with Lemma 2.2 and the second condition of Theorem. Similarly, if there exist n goodfaces among F , F , . . . , F n − , then there are n pairwise non-neighbouring faces of K , given by thevectors f , f , . . . , f n ∈ S n , such that |h w, f i| < βn . Again, this contradicts Lemma 2.2 combinedwith the third condition of Theorem. This completes the proof. (cid:3) Random spherical polytopes
In this Section we derive Theorem 1.2 from Theorem 1.1. In order to do this, we need toestablish several probabilistic lemmas. These results are strongly based on the ideas developed bydifferent authors. We shall rephrase or modify them, according to our needs. We start with a lowerbound on the probabilistic measure of the spherical cap, that was proved in [6].
Lemma 3.1.
Let x ∈ S n and < r < . Then, the probabilistic measure of the spherical cap C ( x, r ) = { y ∈ S n : || x − y || ≤ r } is at least √ π ( n − (cid:16) r √ (cid:17) n − .Proof. If < ϕ ≤ π is such that r = 2 − q − sin ϕ, then by the first part of Corollary 3.2 in [6], it follows that the measure of C ( x, r ) is at least p π ( n −
1) sin n − ϕ. However r = 2 − q − sin ϕ = 2 sin ϕ p − sin ϕ ≤ ϕ and hence the claim follows. (cid:3) In [5] there is an outline of the proof, that for any fixed ε > , the probability that N randompoints on the unit sphere in R n form a N − n − + ε -net tends to , as N → ∞ (see the proof of (1 . in Appendix A). In order to obtain some more explicit estimate, we give a minor modification ofthis argument in the following lemma. Lemma 3.2.
Let N ≥ n n be a positive integer. If x , x , . . . , x N ∈ S n are random points, then theprobability, that these random points do not form a n -net in the unit sphere, is less than N . roof. It is well-known, that for any ε > , there exists a ε -net in the unit sphere of cardinalityat least (cid:0) ε (cid:1) n . Hence, let z , z , . . . , z n be some fixed n -net in the unit sphere. For a fixed ≤ i ≤ n , the probability that each point x , x , . . . , x N is outside the cap C ( z j , n ) , is, byLemma 3.1, at most − p π ( n − (cid:18) √ n (cid:19) n − ! N ≤ (cid:18) − (cid:18) √ n (cid:19) n (cid:19) N . Therefore, the probability that at least one of the caps C ( z j , n ) is empty, is at most n (cid:18) − (cid:18) √ n (cid:19) n (cid:19) N ≤ e · e − Na − n = e − Na − n , where a = 8 √ n . Since N ≥ n n and n ≥ we have that N a − n ≥ √ N .
It is easy to check that for N ≥ n n ≥ the inequality N < √ N is true. Hence, theprobability that at least one of the caps C ( z j , n ) is empty is less than N . It remains to observe,that if each of these caps is non-empty, then the points x i form a n -net in the unit sphere. (cid:3) A convex polytope in R n with N vertices can have as much as O ( N ⌊ n ⌋ ) faces (as an example ofa cyclic polytope shows). Perhaps surprisingly, it turns out, that the expected number of faces of arandom spherical polytope with N vertices is of a much smaller order - asymptotically, with fixed n , it is roughly only of a linear order in N . This kind of result appeared first in [7]. Our goal is togive a uniform bound on the expected number of faces, rather than just the asymptotic order. Todo this, we use the ideas developed in [16]. The next technical lemma is of a crucial importance andis heavily based on Lemma . from [16]. We recall that the classical Gamma function is definedas Γ( x ) = R ∞ t x − e − t dt for x > . Lemma 3.3.
Let α, β > − , c > and B m = Z − (1 − t ) β − (cid:18)Z t − c (1 − s ) α − ds (cid:19) m dt, for an integer m ≥ . Then B m ≤ m − β +1 α +1 β +12 α + 1 (cid:18) α + 1 c (cid:19) β +1 α +1 Γ (cid:18) β + 1 α + 1 (cid:19) . Proof.
We mimic the proof of Lemma . in [16], repeating almost exactly the same argument,but replacing asymptotic relations with upper bounds. We define g m ( u ) = m β − α +1 (cid:18) − (cid:16) − um − α +1 (cid:17) (cid:19) β − (cid:18) − Z − um − α +1 c (1 − s ) α − ds (cid:19) m , for m ≥ and u > . Then, by the change of variables − t = um − α +1 we have B m = m − β +1 α +1 Z m α +1 g m ( u ) du. We shall lower bound the integral Z − um − α +1 (1 − s ) α − ds, or ≤ p ≤ we have Z − p (1 − s ) α − ds = Z p (2 v − v ) α − dv Let F ( p ) = R p (2 v − v ) α − dv . Then F (0) = 0 and F ′ ( p ) = (2 p − p ) α − = p α − (2 − p ) α − ≥ p α − = G ′ ( p ) , where G ( p ) = α +1 p α +12 . Therefore Z − um − α +1 (1 − s ) α − ds ≥ α + 1 u α +12 m − . Combining this estimate with an inequality x ≤ e x we get (cid:18) − Z − um − α +1 c (1 − s ) α − ds (cid:19) m ≤ (cid:18) − cα + 1 u α +12 m − (cid:19) m ≤ exp (cid:18) − cα + 1 u α +12 (cid:19) . Thus g m ( u ) = (cid:16) u − u n − α +1 (cid:17) β − (cid:18) − Z − um − α +1 c (1 − s ) α − ds (cid:19) m ≤ (2 u ) β − exp (cid:18) − cα + 1 u α +12 (cid:19) . Hence B m = m − β +1 α +1 Z m α +1 g m ( u ) du ≤ m − β +1 α +1 Z ∞ (2 u ) β − exp (cid:18) − cα + 1 u α +12 (cid:19) du = m − β +1 α +1 β +12 α + 1 (cid:18) α + 1 c (cid:19) β +1 α +1 Γ (cid:18) β + 1 α + 1 (cid:19) , which finishes the proof. (cid:3) Now we are ready to give an upper bound for the expected number of faces of a random sphericalpolytope with N ≥ n vertices. We again rely on the result from [16], using an integral expressionfor the expected number of faces of a random spherical polytope. Lemma 3.4.
Let N ≥ n be a positive integer. If x , x , . . . , x N ∈ S n are random points, then theexpected value of the number of faces of the convex polytope K = conv { x , x , . . . , x N } is less than n − N .Proof. We start with the case n = 3 . The convex polytope K = conv { x , x , . . . , x N } issimplicial with probability . By the Euler’s polyhedron formula, the number of faces of a threedimensional simplicial polytope with N vertices is equal to N − . Therefore, in the case n = 3 ,our estimate is obviously true.Thus, let us suppose that n ≥ . By Theorem 1.2 in [16] (see also Remark 1.4), the expectednumber of faces of a random spherical polytope with N vertices is equal to c (cid:18) Nn (cid:19) Z − (1 − t ) n − n − (cid:18)Z t − c (1 − s ) n − ds (cid:19) N − n dt, where c = Γ (cid:16) n − n +22 (cid:17) √ π Γ (cid:16) n − n +12 (cid:17) and c = Γ ( n ) √ π Γ ( n − ) . By Lemma 3.3 for β = n − n , α = n − and m = N − n we get that the expected number of faces is at most c (cid:18) Nn (cid:19) ( N − n ) − n · (cid:18) n − · n − c (cid:19) n − · ( n − · n − . e will use the following well-known inequalities on the ratio of Gamma functions. The lower boundappeared in this form in [11] and upper bound in [27]. √ x ≤ Γ ( x + 1)Γ (cid:0) x + (cid:1) ≤ r x + 12 for every x > . (4)Using this, we get c n − (cid:16) n − n +22 (cid:17) ( n − √ π Γ (cid:16) n − n +12 (cid:17) ≤ √ π < . Similarly, we have n − c = ( n − √ π Γ (cid:0) n − (cid:1) Γ (cid:0) n (cid:1) ≤ ( n − √ π p n − . It is now easy to check that for n ≥ , an estimate (cid:18) n − · n − c (cid:19) n − ≤ n − n is true. Since N ≥ n we have also that NN − n ≤ . Therefore c (cid:18) Nn (cid:19) ( N − n ) − n · (cid:18) n − · n − c (cid:19) n − · ( n − · n −
1= 2 c n − · N · ( N − . . . ( N − n + 1)( N − n ) n − · n ( n − · (cid:18) n − · n − c (cid:19) n − < N · n − · n − n = 2 n − N. (cid:3) Remark . Asymptotically speaking, the constant n − in this estimate, is not of an optimalorder. We sacrificed the constant, to be able to get a uniform bound for every N ≥ n randompoints. The optimal constant is of an order or exp( n log n ) , rather than exp( n ) . See Remark 4.4for more details.The last piece of information, that we need to apply Theorem 1.1, in order to prove Theorem1.2, is the expected value of the determinant of n random points in S n . More precisely, we estimatethe ( − ) -moment of the absolute value of the determinant. We use the fact, that the distributionof the random variable | det( x , x , . . . , x n ) | , where x i ∈ S n are random points, is well-known. Lemma 3.6.
Let M n be the expected value of | det( x , x , . . . , x n ) | − , where x i ∈ S n for ≤ i ≤ n are random points. Then, we have M n < e n ( n − .Proof. Let us recall, that a random variable has a
Beta distribution with parameters α , α > ,if its density is given by g ( t ) = Γ( α + α )Γ( α )Γ( α ) t α − (1 − t ) α − , for t ∈ (0 , . It turns out that the random variable det( x , x , . . . , x n ) has the distribution Q n − i =1 β i , n − i , where β α ,α has a Beta distribution with parameters α , α and the variables in theproduct are independent (see [24] and [15]). The ( − ) -moment of the Beta variable with parameters α > , α > is equal to Γ (cid:0) α − (cid:1) Γ( α + α )Γ( α )Γ( α + α − ) . herefore, M n = n − Y i =1 Γ (cid:0) i − (cid:1) Γ (cid:0) n (cid:1) Γ (cid:0) i (cid:1) Γ (cid:0) n − (cid:1) From the inequality (4) we have Γ (cid:0) n (cid:1) Γ (cid:0) n − (cid:1) ≤ (cid:18) n − (cid:19) ≤ (cid:16) n (cid:17) . Similarly, Γ (cid:0) i − (cid:1) Γ (cid:0) i (cid:1) ≤ (cid:0) i (cid:1) i − ≤ (cid:0) i (cid:1) (cid:0) i − (cid:1) , for ≤ i ≤ n − . For i = 1 , we can check by a direct calculation that Γ (cid:0) (cid:1) Γ (cid:0) (cid:1) ≤ (cid:0) (cid:1) . Thus, n − Y i =1 Γ (cid:0) i − (cid:1) Γ (cid:0) i (cid:1) ≤ · (cid:18) ( n − n − (cid:19) − · (cid:18) n − (cid:19) = 2 · (cid:18) ( n − n − (cid:19) − · (cid:18) n − (cid:19) = (cid:18) ( n − n − (cid:19) − · ( n − . Hence, we conclude that M n ≤ (cid:18) n n n ! (cid:19) · ( n − < e n · ( n − , by the Stirling’s approximation formula. This finishes the proof. (cid:3) Finally, we are ready to move to the proof of Theorem 1.2.
Proof of Theorem 1.2.
Let us consider following probability events:(i) Points x , x , . . . , x N do not form a n -net in S n .(ii) The convex polytope B X has more than n N faces.(iii) | det( x i , x i , . . . , x i n ) | ≤ (cid:16) N ( n − (cid:0) Nn (cid:1) e n (cid:17) − for some ≤ i < i ≤ i n ≤ N .(iv) There exist pairwise non-neighbouring faces F , F , . . . , F n of B X , perpendicular to vectors f i ∈ S n (where ≤ i ≤ n ), such that | det( f , f , . . . , f n ) | ≤ (cid:18) N ( n − (cid:0) n N n (cid:1) e n (cid:19) − .We shall prove, that the probability that at least one of these events is true, is less than N . It isenough to show that each of events (1 − has probability at most N and the conditional probabilityof the event (iv), assuming that the event (ii) does not hold, is also at most N . Moreover, we canassume that B X is a simplicial polytope, as this happens with the probability .(i) Our claim follows directly from Lemma 3.2.(ii) Note that, if we pick n -element subset of the set {± x , ± x , . . . , ± x N } , then the probabilitythat this subset forms a face of B X is if this subset contains some symmetric pair of points.Otherwise, this probability is the same positive number for every n -element subset without asymmetric pair of points. Thus, we conclude, that the expected value of the number of facesof B X , is not greater than the expected value of the number of faces of a random sphericalpolytope with N vertices. Therefore, by Lemma 3.4, the expected value of the number of aces of B X is not greater than n − (2 N ) = 2 n N . The desired estimate of the probability isnow a consequence of the Markov’s inequality.(iii) We use Lemma 3.6 and the Markov’s inequality applied for the random variable | det( y , y , . . . , y n ) | − , where y , y , . . . , y n ∈ S n are random points. For any a > we have: P (cid:16) | det( y , y , . . . , y n ) | − ≥ a ( n − e n (cid:17) ≤ a − , which is equivalent to P (cid:18) | det( y , y , . . . , y n ) | ≤ (cid:16) a ( n − e n (cid:17) − (cid:19) ≤ a − . Using this inequality for a = N (cid:0) Nn (cid:1) , with the union bound for all possible choices of n pointsfrom x , x , . . . , x N , we get the desired upper bound of N .(iv) We assume that B X has at most n N faces. It is easy to see, that a set of vectors f i ∈ S n ,corresponding to a set of n pairwise non-neighbouring faces of B X is, in fact, a set of n independent random points in S n , with respect to the uniform distribution (the assumptionthat the faces are non-neighbouring is crucial here). Thus, we can use the Markov’s inequalityand the union bound for at most (cid:0) n N n (cid:1) of all possible choices, exactly as for the previousevent.The result follows now immediately from Theorem 1.1 with α = (cid:18) N ( n − (cid:18) Nn (cid:19) e n (cid:19) − and β = N ( n − (cid:18) n N n (cid:19) e n ! − . (cid:3) Concluding remarks
In the following section we present some remarks related to previous results. We start with thethree-dimensional setting.
Remark . In the three-dimensional case, the uniform estimate given in Theorem 1.2 can beimproved to the form || P || ≥ C N (cid:18) N (cid:19) (cid:18) N − (cid:19) ! − ≥ cN − , for some easily computable constant c > . This follows from the fact, that in the three dimensionalcase, K simplicial with the probability and the number of faces is equal to N − by the Euler’spolyhedron formula. By taking N = 3 we get a first non-trivial lower bound on ρ = ρ H . Betternumerical bound on ρ will be given in the next remark. Remark . It is not hard to prove, that in the three-dimensional case, the condition (1) in Theorem1.1 can be replaced with an assumption, that the volume of K is greater than (here we mean thestandard volume in R ) and length of every edge of K is less than . With a help of a computerprogram, a spherical polytope K , satisfying this conditions was found. Number of vertices of K isequal to and α = 5 . · − , β = 1 . · − . This gives us a better numerical estimate thanin Remark 4.1: ρ > · − . It is rather hard to believe, that this estimate could be close to the true value of ρ . Still, in theclass of spherical polytopes with vertices, it is not necessarily so weak. t turns out that if the dimension is fixed, the polynomial bound, given in terms of N , can beimproved. Remark . For the asymptotics with n fixed and N → ∞ , the estimate given in Theorem 1.2 canbe strengthened to c n,ε N − n − − ε for any ε > and some constant c n,ε , depending on n and ε .Indeed, the expected value of a random variable | det( x , x , . . . , x n ) | − ε is finite for every ε > .This can be easily deduced from the distribution of a random variable det( x , x , . . . , x n ) , given inLemma 3.6. Thus, we can replace the exponent of − , in the right hand sides of the inequalitiesgiven in the properties ( iii ) and ( iv ) in the proof of Theorem 1.2, to the exponent of − − ε , albeitwith some different constant depending on n and ε .As we have already mentioned in the introductory section, an asymptotic lower bound on theconstant p Hn obtained in [18] can be easily recovered with our methods. Remark . By taking N = n n in Theorem 1.2 we obtain a lower bound ρ Hn ≥ exp( − Cn ) s(where C > is an absolute constant). However, as we mentioned in Remark 3.5, the constant n − in Lemma 3.4 is not asymptotically optimal. From Theorem 13 in [3] it follows, that if wetake N = 2 n n , then the expected number of faces of a random spherical polytope with N vertices,does not exceed CK n N = CK n n n for some absolute constant C > and K n is defined as K n = 2 n π n − n ( n − · Γ (cid:16) n − n +22 (cid:17) Γ (cid:16) n − n +12 (cid:17) · Γ (cid:0) n +12 (cid:1) Γ (cid:0) n (cid:1) ! n − . From the inequality (4) it follows that K n ≤ n · ( n − · (cid:16) n (cid:17) n − . We conclude, that the expected number of faces of the symmetric convex hull of n n randompoints in the unit sphere does not exceed n cn , for some constant c > . It means that, by repeatingthe proof of Theorem 1.2 for N = n n , but with number of faces n n n in the second event replacedwith n cn (which is of order exp( cn log n ) , instead of exp( cn ) ), we get exactly the uniform lowerbound of the form n − c n = 1 + exp( − c n log n ) on the norms of all hyperplane projections(for some easily computable constant c ).We must note, that the asymptotic lower bound ρ Hn ≥ exp( − Cn log n ) does not seem to beoptimal. The same is true for the estimate ρ > · − on the three-dimensional constant.Moreover, our results do not give any non-trivial bound on the constant ρ n for n ≥ . Providingsome different example of a class of normed spaces, for which all non-trivial projections/hyperplaneprojections satisfy some explicit uniform lower bound on the norm, is of its own interest, even ifit does not lead to improvement in the global estimates. Considering the importance of randomconstructions in modern functional analysis, it is reasonable to believe, that random polytopescould possibly yield much better bounds on the constants ρ n and ρ Hn . It is likely, that our methodscould be extended to all non-trivial projections, providing some bound on ρ n . However, to obtaina better bound on ρ Hn , it would be probably necessary to use random polytopes with the numberof vertices of a smaller order than n n . Even in the hyperplane setting, this would possibly requiresome completely new ideas. 5. Acknowledgements
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