Lower Bounds for k-distance Approximation

Transcription

1 Lower Bouns for k-distance Approximation Quentin Mérigot March 21, 2013 Abstract Consier a set P of N ranom points on the unit sphere of imension 1, an the symmetrize set S = P ( P). The halving polyheron of S is efine as the convex hull of the set of centrois of N istinct points in S. We prove that after appropriate rescaling this halving polyheron is Hausorff close to the unit ball with high probability, as soon as the number of points grows like Ω( log()). From this result, we euce probabilistic lower bouns on the complexity of approximations of the istance to the empirical measure on the point set by istance-like functions. 1 Introuction The notion of istance to a measure was introuce in orer to exten existing geometric an topological inference results from the usual Hausorff sampling conition to a more probabilistic moel of noise [CCSM11]. Consier a finite subset P of the Eucliean space R an a positive number k in the range {1,..., P }, where P enotes the carinality of P. The istance to the empirical measure on P is given by the following formula: P,k (x) := ( [ 1 min k p 1,...,p k P ]) 1/2 k x p i 2, (1.1) i=1 where the minimum is taken over the sets consisting of k istinct points in P. We will call this function the k-istance to the point set P. Equation (1.1) allows to compute the value of the k-istance at a certain point x easily, using a nearest-neighbor ata structure. On the other han, this formula cannot be use to perform more global computations, such as estimating the Betti numbers of a sublevel set 1 P,k (0,r) = {x R ; P,k (x) r}, not to mention reconstructing a simplicial complex homotopic to this set. There is another representation of the k-istance that allows to perform such global operations. It is computational geometry folklore that P,k can be rewritten as the square root the minimum of a finite number of quaratic functions. More precisely P,k (x) 2 = min p x p 2 +w p, where the minimum 1

2 is taken over the set of centrois of k istinct points in P, an w p is chosen aequately (see 2.2). This implies that sublevel sets of P,k are simply union of balls, an this allows one to compute their homotopy type using weighte alpha-complexes or similar constructions [Ee92]. However, since the number of centrois of k istinct points in P grows exponentially with the number of points, this formulation is not very practical either. Fortunately, many geometric an topological inference result continue to hol if one replaces P,k in the computation by a goo approximation ϕ [CCSM11]. This means that the error ϕ P,k := max R ϕ P,k has to be small enough, an that the approximating function ϕ shoul be istance-like. For the purpose of this work, ϕ is istance-like if there exists a finite set of sites Q R an non-negative weights (w q ) q Q such that ϕ = ϕ w Q, where ϕw Q is efine by ϕ w Q(x) := ( ) 1/2 min q Q x q 2 +w q, w q 0. (1.2) To summarize, in orer to estimate the topology of the sublevel set of the k-istance, it makes sense to try to replace it by a istance-like function that uses much fewer sites. Note that one coul try to fin approximations of the k-istance in a class of functions F ifferent from the class of istance-like functions. It is inee possible that a well chosen class of functions woul prouce more compact approximations of the k-istance an of similar functions. However, changing the class of function woul practically forbi to use these approximations for the purpose of geometric inference, because of the lack of (i) computational topology tools to compute with the sublevel sets of functions in F an (ii) a geometric inference theory aapte to this class of function. Complexity of k-istance The natural formalization of our approximation problem is as follows. Given a finite point set P in R, a number k > 0 an a target approximation error ε, what is the minimum carinality of a weighte point set (Q, w) with non-negative weights such that the approximation error ϕ w Q P,k is boune by ε? We call this carinality the ε- complexity of the k-istance function P,k. When ε is zero, the 0-complexity of the k-istance function P,k is equal to the number of orer-k Voronoi cell of P that have non-empty interior. One can then translate lower-bouns on the number of orer-k Voronoi cells into lower bouns for the 0-complexity of the k-istance. The purpose of this article is to provie lower bouns on ε-complexity of the k-istance function for a non-zero approximation error ε, when P is a ranom point clou on the unit ( 1)-imensional sphere an k = P /2. 2

3 1.1 Prior work Approximation of the k-istance The question of approximating the istance to the measure by a istance-like function with few sites has been originally raise in [GMM12]. In this article, the authors propose an approximation of the k-istance, calle the witnesse k-istance an enote by w P,k, which involves only a linear number of sites. They also give a probabilistic upper boun on the approximation error w P,k P,k uner the hypothesis that the point clou P is obtaine by sampling a l-imensional submanifol of the Eucliean space. The upper boun on the approximation error egrae as the intrinsic imension l of the unerlying submanifol increases. This suggests that approximating the istance to the uniform measure on a point clou rawn from a high-imensional submanifol might be ifficult. It is possible to buil ata structures that allow to compute approximate pointwise values of the k-istance in time that is logarithmic in the number of points but exponential in the ambient imension [HPK12]. The same ata structure can be use to compute generalizations of the k-istance, such as the sum of the pth power to the k-nearest neighbors for an exponent p larger than one. Complexity of orer-k Voronoi As mentione earlier, there exists upper an lower bouns for the number of cells in an orer-k Voronoi iagrams, an those bouns can be translate into bouns on the number of sites that one nees to use in orer to get an exact representation of the k-istance function by a istance-like function. When k is half the carinality of the point clou P, we will speak of halving Voronoi iagram. A halving hyperplane for P is a hyperplane that separates P into two sets with equal carinality. The number of halving hyperplanes yieling ifferent partitions of P is a lower boun on the number of infinite halving Voronoi cells. The best lower boun on the number of halving hyperplanes, that hols for an arbitrary ambient imension, is given by N 1 e Ω( logn), where N is the carinality in the point set [Tót01]. This boun improves on previous lower bouns by several authors, e.g. [ELSS73, EW85, Sei87]. In [BS94], the authors stuy the expectation of the number of k-sets of a point clou P obtaine by sampling inepenent ranom point on the sphere. Recall that a k-set is a subset of k points in P that can be separate from other points in P by a hyperplane; in particular, each k-set correspons to an infinite orer-k Voronoi cell. The authors prove that the expecte number of k-sets in a ranom point clou P on the sphere is upper boune by O( P 1 ). In orer to obtain our lower bouns on the number of sites neee to approximate P,k, we will use the notion of k-set polyheron, originally introuce in [EVW97]. This polyheron is the convex hull of the set of 3

4 centrois of k istinct points in P. The relation between this polyheron an the notion of k-set is that the number of extreme points of the k-set polyheron is equal to the number of k-sets in the point set. When k is half the carinality of P, we will call this polyheron the halving polyheron. 1.2 Contributions The main result of this work concerns the geometry of halving polyhera of finite point sets on the unit sphere. Our theorem shows that even with relatively few points, the halving polyheron of a certain ranom point set S on the unit( 1)-imensional sphere is Hausorff-close to a ball with high probability. The ranom point set S is obtaine by picking N ranom, inepenent an uniformly istribute points p 1,...,p N on the ( 1)-imensional unit sphere, an by letting S = {±p i } 1 i N. The halving polyheron of S is a by efinition a ranom convex polyheron, which we enote by L N. The statement an proof of this theorem are inspire by the main theorem of [AAFM06]. We use the quantity m := E( X u ) ( ) 2 1/2, π where X is a uniformly istribute ranom vector on the unit ( 1)-sphere, an u is an arbitrary unit vector. Note that this quantity turns out not to epen on u. Theorem 3.1. There exists an absolute constants c > 0 such that for every positive number η, the inequality H ( 1 m L N,B(0,1) ) η hols with probability at least 1 2exp [ c (log(1/δ) Nη 2)], where δ = min(η,1/ ). In Section 4, we euce from this theorem a probabilistic lower boun on the ε-complexity of the S,N, where S is the point set efine in the previous paragraph. The exact statement of the lower boun can be foun in Theorem Traces at infinity an k-set polyhera Backgroun: support function The support function of a convex subset K of R is a function x h(k,x) from the unit sphere to R. It is efine by the following formula: h(k,u) := max{x u; x K}. (2.3) 4

5 The application that maps a convex set to its support function on the sphere satisfies the following isometry property: h(k,.) h(l,.),s 1 = H (K,L). (2.4) where f,s 1 = max S 1 f is the infinity norm on the unit sphere, an where H enotes the Hausorff istance. The proof of this equality is given in Theorem in [Sch93]. All the elementary facts about support functions that we will nee can be foun in the first chapter of this book. 2.1 Traces at infinity of istance-like functions. We call istance-like a non-negative function whose square can be written as the minimum of a family of unit parabolois x q 2 +w q, with w q 0. Note that this efinition is equivalent to the one given in [CCSM11], thanks to the remark following Proposition 3.1 in this article. Given a finite subset Q of the Eucliean space an non-negative weights w, we let ϕ := ϕ w Q be the istance-like function efine by (1.2). We call trace at infinity of ϕ an enote by K(ϕ) the convex polyhera obtaine by taking the convex hull of the set of sites Q use to efine ϕ. The name trace at infinity is explaine by the following asymptotic evelopment of the values of ϕ, along a unit-spee ray starting at the origin, i.e. γ t := tu with u = 1: ϕ(γ t ) = min q Q = min q Q t ( tu q 2 +w q ) 1/2 ( 1 2 t u q + w q + q 2 t 2 ) 1/2 = t maxu q +O(1/t) (2.5) q Q = t h(k(ϕ),u)+o(1/t) (2.6) The last equality follows from the fact that the support function of the convex hull of a set Q is given by max q Q u q. Using these computations, we get the following lemma. This lemma implies in particular that if two functions ϕ := ϕ w Q an ψ := ϕw P coincie on R, then K(ϕ) = K(ψ). Lemma 2.1. Given two istance-like functions ϕ := ϕ w Q an ψ := ϕv P as in (1.2), H (K(ϕ),K(ψ)) ϕ ψ. (2.7) Proof. The asymptotic evelopments for ϕ an ψ given in equation (2.6) imply the following inequality for every unit irectionu: h(k(ϕ),u) h(k(ψ),u) ϕ ψ. With Equation (2.4), this yiels the esire boun on the Hausorff istance between K(ϕ) an K(ψ). 5

6 The power iagram of a weighte point set (Q, w) is a ecomposition of the space into convex polyhera, one per point in Q, efine by Pow w Q(q) = {x R ; p Q, x q 2 +w q x p 2 +w p }. The following lemma shows the relation between the vertices of the trace at infinity of ϕ w Q an unboune cells in the power iagram of (Q,w). Lemma 2.2. Consier a finite weighte point set (Q,w): (i) if the power cell of a point q in Q is unboune, then q lies on the bounary of the polyheron K(ϕ w Q ); (i) conversely, if q is an extreme point in K(ϕ w Q ), then the power cell of q is unboune. Note that there might exist point q that lie on the bounary of the trace at infinity an whose power cell is empty. For instance, consier Q = {q 1,q 0,q 1 } in R 2 with q i = (0,i), an weights w 1 = w 1 = 0 an w 0 > 1. Then q 0 lies on the bounary of the convex hull while its power cell is empty. Proof. Being convex, the power cell of a point q is unboune if an only it contains a ray γ t := r + tu, where u is a unit vector. By efinition of the power cell, we have q γ t 2 +w q p γ t 2 +w p for all point p in the set Q. Expaning both sies expressions an simplifying, we get: q r 2 p r 2 +2tu (p q)+w q w p 0 This inequality hols for t +, thus implying u p u q. Thus, q lies on the bounary of K(ϕ), an u is an exterior normal vector to K(ϕ) at q. Conversely, if q is an extreme point of the convex polyheron K(ϕ), there must exist a unit vector u, such that u (p q) ε < 0 for any point p in Q istinct from q. Tracing back the above inequalities, an using the strict boun, one can show that for any point r, the ray γ t := r + tu belongs to the power cell of q for t large enough. 2.2 Trace at infinity of the k-istance. Given a set of points P in R an an integer k between one an P, the k-istance to P is efine by equation (1.1). The fact that this function is istance-like can be seen in the following equivalent formulation, whose proof can be foun for instance in Proposition 3.1 of [GMM12]: P,k (x) = ( min p ) 1/2 k x p 2 +w p, (2.8) i=1 where the minimum is taken over the centrois p ofk istinct points in P, i.e. p = 1 k 1 i k p i an where the weight is given by w p := 1 k 1 i k p p i 2. 6

7 Definition 2.1. We enote Kk (P) the convex hull of the set of centrois of k istinct points in P. The support function of this polyheron is given by the following formula: h(kk 1 (P),u) := max p 1,...,p k P k k u p i (2.9) This polyheron has been introuce first in [EVW97] uner the name of k-set polyheron of the point set P. Equations (2.8) (2.9) above imply that this polyheron is the trace at infinity of the k-istance to P. Moreover, the number of infinite orer-k Voronoi of P is at least equal to the number of extreme points in K k (P). i=1 Halving istance When the number of points in P is even an k is equal to half this number, we rename the k-istance the halving istance. Similarly, we will refer to the orer-k Voronoi iagram as the halving Voronoi iagram an to the k-set polyheron as the halving polyheron. 3 Approximating the sphere by halving polyhera In this section, we consier a family of ranom polyhera constructe as halving polyheron of symmetric point sets on the unit sphere. More precisely, we efine: Definition 3.1. Given a set P of N points on the unit ( 1)-sphere, we efine L N (P) as the halving polyheron of the symmetrization of P, i.e. L N(P) := K N(P ( P)) We obtain a (ranom) convex polyheron L N := L N (P), by letting P be an inepenent ranom sampling of N points on the unit ( 1)-sphere. Our main theorem consists in a lower boun on the probability of the halving polyheron L N to be Hausorff-close to a ball centere at the origin. Theorem 3.1. There exists an absolute constants c > 0 such that for every positive number η, the inequality ( ) 1 H L m N,B(0,1) η hols with probability at least 1 2exp [ c (log(1/δ) Nη 2)], where δ = min(η,1/ ). The proof of this theorem is postpone to Section 5. As a first corollary, one can show that the ranom polyheron L N is approximately roun with high probability as soon as the carinality of the set of points sample on the sphere grows faster than log, where is the ambient imension. 7

8 Corollary 3.2. For any κ > 0, there is a constant C κ such that for η > 0, 1/η 2 an N (log()+c η 2 κ ) the following inequality ( ) 1 H L m N,B(0,1) η hols with probability at least 1 exp( κ). Proof. From Theorem 3.1 the probability boun in the statement hols if κ c(log(1/δ) Nη 2 ). Since δ is equal to 1/, this is the case if Nη 2 ( log()+ 2κ ) c. 4 Application: approximation of istance-to-measures The results of the previous section can be use to obtain a probabilistic statement on the complexity of the halving istance to a ranom point set on a high-imensional sphere. We call ε-complexity of a istance-like function ϕ : R R the minimum number of sites that one nees in orer to be able to construct a istance-like function ψ such that the infinity norm ϕ ψ is at most ε, i.e. { N(ϕ,ε) := min Q ; ϕ ϕ w } Q ε, ϕ w Q as in (1.2). The following theorem provies a probabilistic lower boun on the ε-complexity of a family of istance-like functions. Theorem 4.1. For any constant κ > 0 there exists a constant C(κ) such that the following hol. Let η > 0, 1/η 2, an S be the symmetrization of an ranom point clou of carinality N = (log() + C η 2 κ ) on the unit sphere. Then, the inequality N ( S,N,m η) 2 ( N 64(log()+C κ ) ) 1 4 hols with probability at least 1 exp( κ). Taking κ = 1 an = 1/η 2, this implies: Corollary 4.2. There exists a sequence of point clous S of carinality 2 (log()+c 1 ) on the sphere S 1 such that N ( S, 1 2 S, 2/π ) 2 ( )

9 Proofof Theorem 4.1. The halving polyheron L N (S) is by efinition equal to the trace at infinity of the halving istance K( S,N ). Therefore, we can apply Corollary 3.2: there exist a constant C κ such that for a ranom point set S istribute as in the statement of the theorem, the following inequality hols: ( ) 1 H K( S,N ),B(0,1) η. (4.10) m with probability at least 1 exp( κ). We now consier a eterministic point set S on the sphere that satisfies the above inequality, an we consier a istance-like function ψ : x min q Q x q 2 + w q such that the approximation error ψ S,N is boune by m η. By efinition of the complexity of S,N, our goal is to prove a lower boun on the carinality of the point set Q. Using the triangular inequality for the Hausorff istance first, an then Lemma 2.1.(ii), we get the following inequalities: ( ) 1 H K(ψ),B(0,1) m 1 ( ) 1 H (K(ψ),K( S,N ))+ H K( S,N ),B(0,1) m m 1 m ψ S,N +η 2η Now, recall that K(ψ) is equal to the convex hull of the point set Q. If we efine R as the rescale set {m 1 q; q Q}, the above inequality reas H (conv(r),b(0,1)) 2η. We have thus constructe a polyheron that is within Hausorff istance 2η of the -imensional unit ball. The last remark of [BI75] gives the following lower boun on the number of vertices of such a polyheron: Q = R 2 (8η) 1 2 To conclue the proof, we simply replace η by its expression in term of the number of points N an the imension, i.e. η 2 := (log() + C κ )/N to obtain Q 2 ( N 64(log()+C κ ) 5 Proof of the main theorem ) 1 4 We start this section by showing a simple expression for the support function of the halving polyhera of a symmetric point set. For a point set P, an N := P, one has: h(l N(P),u) = p u. (5.11) p P 9

10 Proofof Equation (5.11). Let S = P { P} an recall that by efinition, the support function h(l N (P),u) is equal to max N i=1 u p i, where the maximum is taken on sets of N istinct points in S. Choosing ε(p) = ±1 such that for every point in p in P, ε(p)u p 0, one easily sees that this maximum is attaine for {p 1,...,p N } = {ε(p)p;p P}. This computation motivates the following efinition. Definition 5.1. We letλ 1,u be the measure on[0,1] given by the istribution of u X where X is a uniformly istribute ranom vector on the ( 1)- imensional unit sphere, an u is a fixe unit vector. This measure turns out not to epen on u, so we will enote it by λ 1. The ranom polyheron L N is efine as L N (P) where the point set P is obtaine by rawing N inepenent points on the unit sphere. Equation (5.11) implies that for any unit vector u the istribution of values of h(l N,u) is given by the formula: 1 N N Y i, (5.12) i=1 where the (Y i ) are N inepenent ranom variables with istribution λ 1. Lemma 5.1. The measure λ 1 has the following properties: (i) λ 1 is absolutely continuous with respect to the Lebesgue measure, with ensity f (t) := c (1 t 2 ) 2 2 the constant c being chosen so that f is the ensity of a probability 1 measure, i.e. c 0 (1 t2 ) 2 2 t = 1. (ii) the mean of λ 1 is given by m := c. Moreover, m is equivalent to 2/(π) as. (iii) The variance σ 2 of λ 1 is equivalent to (1 2/π)/ as. Proof. (i) Using Pythagoras theorem, one checks that the intersection of the hyperplane {x R ; u x = t} with the unit sphere is a ( 2)-imensional sphere with square raius (1 t 2 ). This implies the formula for f, for a certain constant c. To compute this constant one uses the fact that λ 1 has unit mass, i.e. 1 c (1 t 2 ) 2 2 t = 1. Note that a formula of Wallis asserts that lim 0 1 (1 t 2 ) 2 = π/

11 This implies that c 2/π. (ii) The function t tf (t) amits an explicit primitive: g (t) := c (1 t2 ) 2, so that m = g (1) g (0) is as in the statement of the lemma. Thus, we get m 2/(π). (iii) Integrating the following inequality between 0 an 1 t 2 (1 t 2 ) 2 2 = (1 t 2 ) 2 2 (1 t 2 )(1 t 2 ) 2 2, one gets the following formula for the secon moment of λ 1 : c 1 Moreover, an integration by part gives c t 2 (1 t 2 ) 2 2 t = 1 c c +2. t 2 (1 t 2 ) 2 2 t = c 1 0 (1 t 2 ) 1 2 t = c c +2 These two equalities imply that c /c +2 = /(+1), an using the formula for the mean given above, we get: σ 2 = 1 c /π. Lemma 5.2. There exists a universal constant c > 0 such that for any imension, any N > 0 an any set of irections U in S 1, one has ( ) P h(l N,u) m ηm 2 U exp ( c Nη 2). max u U Proof. Consier N ranom variables Y 1,...,Y N with istribution λ 1. These ranom variable are boune by 1 an their variance is σ 2. Applying Bernstein s inequality gives: ) 1 N ( ) Nε 2 P( Y i m ε 2exp N 2σ 2 +2ε/3 i=1 This implies that for a fixe irection u, ( ) h(l Nη P N,u) m ηm 2exp( 2 m 2 ) 2σ 2 +2ηm /3 Rescaling everything by a certain constant κ > 0, one gets ( ) ( ) h(l h(κl P N,u) m ηm = P N,u) κm κηm Nκ 2exp( 2 η 2 m 2 ) 2κ 2 σ 2 +2ηκm /3 11

12 Letting κ go to infinity, we obtain the following boun ( ) ( h(l P N,u) m ηm 2exp 1 ) 2 Nη2m2 σ 2 This gives the esire estimate for a single irection using the two estimates m = O( 1/2 ) an σ 2 = O( 1 ). The conclusion of the lemma is obtaine by a simple application of the union boun. Lemma 5.3. If K is containe in the ball B(0,r), the support function h(k,.) is r-lipschitz. Proof. Consier u in the unit sphere, an x in K such that h(k,u) = u x. For any vector v in the unit sphere, h(k,v) = maxv y v x = u x+(v u) x y K Swapping u an v gives the Lipschitz boun. h(k,u) u v x h(k,u) r u v. A subset U of the unit sphere S 1 is calle a δ-sample or a δ-covering if the union of the Eucliean balls of raius δ centere at points of U cover the unit sphere. Lemma 5.4. Consier a convex set K containe in the unit ball B(0, 1), an two numbers λ, η (0, 1). Moreover, suppose max h(k,u) λ ηλ, (5.13) u U where U is a δ-sample of the unit sphere, with δ := min(λ,η). Then, the Hausorff istance between 1 λk an the unit ball is at most 5η. Proof. Note that, almost by efinition, a convex set K is inclue in the ball B(0,r) if an only if its support function satisfies h(k,.) r. Assuming that the convex set K is containe in some ball B(0,r), we have: h(k,.) λ = max h(k,v) λ v S 1 max v S 1 [ min u U h(k,u) h(k,v) + h(k,u) λ rmin(λ,η)+ηλ. (5.14) The first inequality is obtaine by applying the triangle inequality, while the secon one follows from the Lipschitz estimation of Lemma 5.3, the fact that U is a δ-sample of the unit sphere an from Equation Applying ] 12

13 Inequality (5.14) with r = 1, η < 1 an using the triangle inequality we get h(k,.) λ 2λ. This implies that K is containe in the sphere B(0,3λ) an allows us to apply the same inequality (5.14) again with the smaller raius r = 3λ, implying h(k,.) λ 3λη +2λη = 5λη. Diviing this last inequality by λ, an using Equation (2.4) implies the conclusion of the Lemma. Proof of Theorem 3.1 Consier δ = min(η,m ), an let U be a δ-sample of the unit sphere with minimal carinality. The carinality of such a sample is boune by U (const/δ), where the constant is absolute. Applying Lemma 5.4 first an then Lemma 5.2 gives us the following inequalities: P ( ( ) 1 H K m N,B(0,1) ) ( 5η P max u U ) h(l N,u) m ηm 2 U exp ( c Nη 2). We conclue the proof by applying the upper boun on the carinality of the δ-sample U state above, an by using the equivalent m 2/(π). 6 Extension to other values of k N It is possible to exten some of the results to cases where ratio k/n is ifferent from one half. Let us consier the ranom polytopem N,k = K k (P), where P is a set of N ranom points sample uniformly an inepenently on the ( 1)-imensional unit sphere, an k is between 1 an N. Lemma 5.2 can be partially extene to this more general family of ranom polytopes. Note however that the statement below oes not provie any estimate for the raius r(,n,k) as the number of points N grows to infinity with k/n remaining constant. In particular, it is not precise enough to generalize the lower bouns of Theorem 3.1. Lemma 6.1. For any imension, an any numbers N an k, there exist a value r := r(,n,k) such that for any set of irections U in S 1, an N > 0 one has ( P max u U ) h(mn,k,u) r ηr 2 U exp ( 1 k 2 ) 4N η2 r 2. This lemma follows from a version of Chernoff s inequality aapte to Lipschitz functions. Consier a function F from the cube [ 1,1] N to R, 13

14 which is α-lipschitz with respect to the l 1 norm on the cube. Then, for any family Y 1,...,Y N of i.i.. ranom variables taking values in [ 1,1] one has: ( ) ε 2 P( F(Y 1,...,Y N ) EF ε) 2exp 4α 2. (6.15) N Proofof Lemma 6.1. We consier the map FN k from the cube [ 1,1]N to R efine by: FN k 1 k (x) := max x ij ; 1 i 1 <... < i k N k, j=1 where x i enotes the ith coorinate of x. We also consier the measure µ obtaine by pushing forwar the ( 1)-area measure on the unit sphere of R by the map x u x, for some irection u in the unit sphere. As in the halving case (cf (5.12)), the istribution of h(mn,k,u) is given by the istribution of FN k(y 1,...,Y N ), where Y 1,...Y N are i.i. ranom variables with istribution µ. Moreover, the map FN k is 1 k -Lipschitz with respect to the l1 norm on the cube [ 1,1] N. Inee, given two points x,y in [ 1,1] N an a set of inices i 1 <... < i k corresponing to the maximum in the efinition of FN k (x), one has kf k N (x) = x i x ik y i y ik + x y l 1 kf k N (y)+ x y l 1 Thus, we can apply Chernoff s inequality (6.15): ) F k P( N (Y 1,...,Y N ) EFN k ε 2exp ( ε2 k 2 ) 4N where m = k/n. The Lemma follows by setting r := EFN k, ε = ηr in the equation an by using the union boun. Acknowlegements This work was partially fune by French ANR grant GIGA ANR-09-BLAN References [AAFM06] S. Artstein-Avian, O. Frielan, an V. Milman, Geometric applications of Chernoff-type estimates an a zigzag approximation for balls, Proceeings of the American Mathematical Society 134 (2006), no. 6,

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