THE COVERING NUMBERS AND LOW M -ESTIMATE FOR QUASI-CONVEX BODIES

Transcription

1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 17, Number 5, Pages S ) Article electronically ublished on January 9, 1999 THE COVERING NUMBERS AND LOW M -ESTIMATE FOR QUASI-CONVEX BODIES A. E. LITVAK, V. D. MILMAN, AND A. PAJOR Communicated by Dale E. Alsach) Abstract. This article gives estimates on the covering numbers and diameters of random roortional sections and rojections of quasi-convex bodies in. These results were known for the convex case and layed an essential role in the develoment of the theory. Because duality relations cannot be alied in the quasi-convex setting, new ingredients were introduced that give new understanding for the convex case as well. 1. Introduction and notation Let be a Euclidean norm on and let D be the ellisoid associated to this norm. Denote by σ the normalized rotationally invariant measure on the Euclidean shere S n 1. For any star-body K in define M K = x dσx), where S n 1 x is the gauge of K. Let MK be M K 0,whereK0 is the olar of K. For any subsets K 1,K of denote by NK 1,K ) the smallest number N such that there are N oints y 1,..., y N in K 1 such that N K 1 y i + K ). Recall that a body K is called quasi-convex if there is a constant c such that K + K ck, andgivena 0, 1] a body K is called -convex if for any λ, µ > 0 satisfying λ + µ =1andanyointsx, y K the oint λx + µy belongs to K. Note that for the gauge = K associated with the quasi-convex -convex) body K the following inequality holds for any x, y : x + y c max{ x, y } x + y x + y ). In articular, every -convex body K is also quasi-convex and K + K 1/ K.A more delicate result is that for every quasi-convex body K K + K ck) there exists a q-convex body K 0 such that K K 0 ck, where 1/q =c. This is the Aoki-Rolewicz theorem [KPR], [R], see also [K],.47). In this note by a body we always mean a comact star-body, i.e. a body K satisfying tk K for all t [0, 1]. Received by the editors Setember 19, 1996 and, in revised form, June 14, Mathematics Subject Classification. Primary 5C17; Secondary 46B07, 5A1, 5A30. This research was done while the authors visited MSRI; we thank the Institute for its hositality. The first and second authors research was artially suorted by BSF c 1999 American Mathematical Society

2 1500 A. E. LITVAK, V. D. MILMAN, AND A. PAJOR Let us recall the so-called low M -estimate result. Theorem 1. Let λ 0, 1) and n be large enough. Let K be a centrally-symmetric convex body in and let be the gauge of K. Then there exists a subsace E of, )such that dim E =[λn] and for any x E the inequality x fλ) MK x holds for some function fλ), 0 <λ<1. Remark. An inequality of this tye was first roved in [M1] with very oor deendence on λ andthenimrovedin[m]tofλ)=c1 λ). It was later shown [PT]) that one can take fλ) =C 1 λfor different roofs see [M3] and [G]). By duality this theorem is equivalent to the following theorem. Theorem 1. Let λ 0, 1) and n be large enough. For every centrally-symmetric convex body K in there exists an orthogonal rojection P of rank [λn] such that PD M K fλ) PK. Theorem 1 was one of the central ingredients in the roof of several recent results of Local Theory, e.g. the Quotient of Subsace Theorem [M1]) and the Reverse Brunn-Minkowski inequality of the second named author see, e.g. [MS] or [P]). Both these results were later extended to a -normed setting in [GK] and [BBP]. The roofs have essentially used corresonding convex results and some kind of interolation. However, the main technical tool in the roof of these convex results, Theorem 1, was a urely convex statement. Let us also note an extension of Dvoretzky s theorem to the quasi-convex setting by Dilworth [D]). In this note we will extend Theorem 1 and Theorem 1 to quasi-convex, not necessarily centrally-symmetric bodies. Since duality arguments cannot be alied to a non-convex body, these two theorems become different statements. Also MK should be substituted by an aroriate quantity not involving duality. Note that by avoiding the use of the convexity assumtion we in fact also simlified the roof for the convex case.. Main results The following theorem is an extension of Theorem 1. Theorem. Let λ 0, 1) and n be large enough n >c/1 λ) ). For any -convex body K in there exists an orthogonal rojection P of rank [λn] such that A M K PD PK, 1 λ) 1+1/ where A = const ln/). Remark. To areciate the strength of this inequality aly it to the standard simlex S inscribed in D. ThenM S n log n and therefore for every λ<1there are λn-dimensional rojections containing a Euclidean ball of radius fλ)/ n log n. AtthesametimeScontains only a ball of radius 1/n. In fact, using this theorem for S rd for some secial value r, we can eliminate the logarithmic factor and obtain the existence of λn-dimensional rojections containing a Euclidean ball

3 LOW M -ESTIMATE FOR QUASI-CONVEX BODIES 1501 of radius f 1 λ)/ n. Another examle is the -convex simlex, S, defined for 0, 1) as a -convex hull of extremum oints of S, i.e. { n+1 } n+1 S = λ i x i ; λ i 0and λ i 1, where {x i } n+1 = extr S. Then Theorem gives us the existence of λn-dimensional n log n ; however S contains rojections containing a Euclidean ball of radius fλ,) n 1/ only a ball of radius 1/n 1/. The roof of Theorem is based on the next three lemmas. The first one was roved by W.B. Johnson and J. Lindenstrauss in [JL]. The second one was roved in [PT] for centrally-symmetric convex bodies and is the dual form of Sudakov s minoration theorem. Lemma 1. There is an absolute constant c such that if ε> c/k and N e εk/c, then for any set of oints y 1,..., y N and any orthogonal rojection P of rank k, µ {U O n j:a1 ε) k/n y j PUy j 1 + ε) ) k/n y j } > 0, where µ is the Haar robability measure on O n and A is an absolute constant. Lemma. Let K beabodysuchthatk+k ak. Then ND, tk) e 8naMK/t). Proof. M. Talagrand gave a direct simle roof of this lemma for the convex case [LT],. 8-83). One can check that his roof does not use symmetry and convexity of the body and roduces the estimate ND, tb) e namb/t) for every body B, such that B B ab. Now for a body K, satisfying K + K ak, denote B = K K. Then B B ab and M B M K,since x B =max x K, x K ) x K + x K. Thus ND, tk) ND, tb) e namk/t). Lemma 3. Let B be a body, K a -convex body, r 0, 1), {x i } rb and B 1 xi + K). ThenB t r K,wheret r =. 1 r ) 1/ Proof. Let t r be the smallest t>0 for which B tk. Then, obviously we have t r =max{ x K x B}. SinceB x i +K), for any oint x in B there are oints x 0 in rb and y in K such that x = x 0 + y. Then by maximality of t r and -convexity of K we have t r r t r + 1. This roves the lemma. Proof of Theorem. Any -convex body K satisfies K + K ak with a = 1/. ByLemmawehave N=ND, tk) ex 3+/ nm K /t) ), i.e. there exist oints x 1,..., x N in D, such that N D x i + tk).

4 150 A. E. LITVAK, V. D. MILMAN, AND A. PAJOR Denote c = 3+/.Lettand ε satisfy c n MK t ) ε k c and ε> c/k for c the constant from Lemma 1. Choose ε = 1 λ λ. Alying Lemma 1 we obtain that there exists an orthogonal rojection P of rank k such that PD k Px i +tp K) and Px i 1 + ε) n x i. Let λ = k/n. Denoter=1+ε) λ.sincekis -convex,lemma3givesus cc M K PD tt r PK for t = ε λ Then for n large enough we get and ε > c λn, r < 1. for A = const ln/) PD A M K PK, 1 λ) 1+1/. This comletes the roof. Theorem can be also formulated in a global form. Theorem. Let K be a -convex body in. Then there is an orthogonal oerator U such that D A M KK + UK), where A = const ln/). This theorem can be roved indeendently, but we show how it follows from Theorem. Proof of Theorem. First, let us assume that K is a symmetric body. It follows from the roof of Theorem that actually the measure of such rojections is large. So we can choose two orthogonal subsaces E 1,E of such that dim E 1 = [n/], dim E =[n+1)/] and P i D A M KP i K, where P i is the rojection on the sace E i i =1,). Denote by I = id = P 1 +P and U = P 1 P.SoP 1 = I+U and P = I U.ThenUisan orthogonal oerator and for any x D we have x = P 1 x + P x A M K I + U A K + K M K + A UK UK M K ) ) K + A I U M K K = A M K K +UK). This roves Theorem for symmetric bodies. In the general case we need to aly the same trick as in the roof of Lemma. Denote B = K K. Then B is a

5 LOW M -ESTIMATE FOR QUASI-CONVEX BODIES 1503 symmetric -convex body so, by the first art of the roof, there is an orthogonal oerator U such that D A M BB + UB). Since B K and M B M K see roof of Lemma ), we get the result. Let us comlement Lemma by mentioning how the covering number NK, td) can be estimated. In the convex case this estimate is given by Sudakov s inequality [S]), in terms of the quantity MK. More recisely, if K is a centrally-symmetric convex body, then NK, td) e cnm K /t). Of course, using duality for a non-convex setting leads to a weak result, and we suggest below a substitute for the quantity MK. For two quasi-convex bodies K, B define the following number MK, B) = 1 x B dx, K where K is the volume of K, and x B is the gauge of B. Such numbers are considered in [MP1], [MP] and [BMMP]. Lemma 4. Let K be a -convex body and let B be a body. Assume B B ab. Then NK, tb) e cn/)amk,b)/t), where c is an absolute constant. Proof. We follow the idea of M. Talagrand of estimating the covering numbers in the case K = D [LT],. 8-83, see also [BLM] Proosition 4.). Denote the gauge of K by and the gauge of B by B. Define the measure µ by dµ = 1 A e x dx, where A is chosen so that dµ =1. Let L = x B dµ. Then µ{ x B L} 1/. Let x 1,x,... be a maximal set of oints in K such that x i x j B t. So the sets x i + t ab have mutually disjoint interiors. Let y i = ab t x i for some b. Then, by -convexity of K and convexity of the function e t,wehave µ{y i +bb} = 1 e x+yi dx A 1 A bb bb = 1 A e yi K e x + y i ) dx Choose b = L. Then µ{bb} 1/ and, hence, bb e x dx e ba/t) µ{bb}. NK, tb) e al/t).

6 1504 A. E. LITVAK, V. D. MILMAN, AND A. PAJOR Now comute L. First, the normalization constant A is equal to A = e x dx = e t ) dtdx = t 1 e t = x 1 dx x 0 t +n 1 e t dt = K Γ 0 x t 1+ n ), where Γ is the gamma-function. The remaining integral is x B e x dx = x B e t ) dtdx = t 1 e t = x 1 x B dx Using Stirling s formula we get This roves the lemma. 0 x t +n e t dt = K MK, B) Γ L ) 1/ n MK, B). 0 x t dxdt 1+ n+1 ) x B dxdt Remark. An analogous lemma for a -smooth 1 ) body K and a convex centrally-symmetric body B was announced in [MP]. Of course, the roof holds for all >0 and every quasi-convex centrally-symmetric body B. More recisely, the following lemma holds. Lemma 4. Let K and B be bodies. Let B B ab and assume that for some >0there is a constant c which deends only on and the body K, such that x + y K + x y K x K +c y K ) for all x, y Rn. Then NK, tb) e cnc/)amk,b)/t), where c is an absolute constant. Lemma 4 is an extension of Lemma in the symmetric case. Indeed, since Euclidean sace is a -smooth sace, then in the case where K = D is an ellisoid, we have c D) = 1. By direct comutation, MD, B) = n n+1 M B.Thus, ND, tb) e cn)mb/t). Define the following characteristic of K: M K = 1 x dx, K where = D is the Euclidean norm associated to D. Lemma 4 shows that for a -convex body K NK, td) e cn/) M K/t). Theorem 3 follows from this estimate by arguments similar to those in [MPi]. K.

7 LOW M -ESTIMATE FOR QUASI-CONVEX BODIES 1505 Theorem 3. Let λ 0, 1) and n be large enough. Let K be a -convex body in and the gauge of K. Then there exists a subsace E of, )such that dim E =[λn] and for any x E the following inequality holds: 1 λ)1/+1/ x x, a MK where a deends on only more recisely a = const ln/) ). Proof. By Lemma 4 there are oints x 1,..., x N in K, such that N<e cn M K/t) and for any x K there exists some x i such that x x i <t. ByLemma1there exists an orthogonal rojection P on a subsace of dimension [δn] such that if t and ε satisfy ) MK c n < ε δn c and ε> t c δn we have b x i := 1 ε)a δ x i Px i 1 + ε) δ x i for every x i.lete=kerp.thendime=λn, whereλ=1 δ.takexin K E. There is an x i such that x x i <t. Hence x x x i + x i t+ Px i b = t+ Px x i) b t+ x x i b t1 + 1 b ) Therefore for n large enough and ) 1/ const c t = MK ε δ we get const t 1 ε) δ. x const ε 1 ε)δ 1/+1/ x. c 1/ MK To obtain our result take ε, say,equalto1/. As was noted in [MP] in some cases M K << MK and then Theorem 3 gives better estimates than Theorem 1 even for a convex body in some range of λ). As an examle, if K = Bl1 n), then M K c n 1/, but MK c n 1/ log n) 1/ for some absolute constant c. 3. Additional remarks In fact, the roof of Theorem shows a more general fact. Fact. Let D be an ellisoid and K a -convex body. Let ND, K) e αn. For an integer 1 k n write λ = k/n. Then for some absolute constant c and γ = c α, k γ n, 1 γ) n)

8 1506 A. E. LITVAK, V. D. MILMAN, AND A. PAJOR there exists an orthogonal rojection P of rank k such that where c 1 is an absolute constant. c 1 1 λ)/) 1/ PD PK, In terms of entroy numbers this means that 1 ) 1/ k/n)/ c 1 PD PK, e k D, K) where e k D, K) = inf{ε >0 ND, εk) k 1 }. It is worthwhile to oint out that Theorem can be obtained from this result. We thank E. Gluskin for his remarks on the first draft of this note. References [BBP] J. Bastero, J. Bernués and A. Peña, An extension of Milman s reverse Brunn-Minkowski inequality, GAFA ), MR 96g:606 [BLM] J. Bourgain, J. Lindenstrauss, V. Milman, Aroximation of zonoids by zonotoes. Acta Math ), no. 1-, MR 90g:4600 [BMMP] J. Bourgain, M. Meyer, V. Milman, A. Pajor, On a geometric inequality. Geometric asects of functional analysis 1986/87), 71 8, Lecture Notes in Math., 1317, Sringer, Berlin-New York, MR 89h:4601 [D] S.J. Dilworth, The dimension of Euclidean subsaces of quasi-normed saces, Math. Proc. Camb. Phil. Soc., 97, , MR 86b:46003 [G] Y. Gordon, On Milman s inequality and random subsaces which escae through a mesh in. Geometric asects of functional analysis 1986/87), , Lecture Notes in Math., 1317, Sringer, Berlin-New York, MR 90b:46036 [GK] Y. Gordon, N.J. Kalton, Local structure theory for quasi-normed saces, Bull. Sci. Math., 118, , MR 96c:46007 [JL] W.B. Johnson, J. Lindenstrauss, Extensions of Lischitz maings into a Hilbert sace. Conference in modern analysis and robability New Haven, Conn., 198), MR 86a:46018 [KPR] N.J. Kalton, N.T. Peck, J.W. Roberts, An F -sace samler, London Mathematical Society Lecture Note Series, 89, Cambridge University Press, Cambridge-New York, MR 87c:4600 [K] H. König, Eigenvalue Distribution of Comact Oerators, Birkhäuser, MR 88j:4701 [LT] M. Ledoux, M. Talagrand, Probability in Banach saces, Sringer-Verlag, Berlin Heidelberg, MR 93c:60001 [M1] V.D. Milman, Almost Euclidean quotient saces of subsaces of a finite-dimensional normed sace. Proc. Amer. Math. Soc ), no. 3, MR 87e:4606 [M] V.D. Milman, Random subsaces of roortional dimension of finite-dimensional normed saces: aroach through the isoerimetric inequality. Banach saces Columbia, Mo., 1984), , Lecture Notes in Math., 1166, Sringer, Berlin-New York, MR 87j:46037b [M3] V.D. Milman, AnoteonalowM -estimate. Geometry of Banach saces Strobl, 1989), 19 9, London Math. Soc. Lecture Note Ser., 158, Cambridge Univ. Press, Cambridge, MR 9e:46018 [MP1] V.D. Milman, A. Pajor, Isotroic osition and inertia ellisoids and zonoids of the unit ball of a normed n-dimensional sace. Geometric asects of functional analysis ), , Lecture Notes in Math., 1376, Sringer, Berlin-New York, MR 90g:5003 [MP] V. Milman, A. Pajor, Cas limites dans des inégalités du tye de Khinchine et alications géométriques. French) [Limit cases of Khinchin-tye inequalities and some geometric alications] C. R. Acad. Sci. Paris Sér. I Math ), no. 4, MR 90d:5018

9 LOW M -ESTIMATE FOR QUASI-CONVEX BODIES 1507 [MPi] V. Milman, G. Pisier, Banach saces with a weak cotye roerty, Isr.J.Math., ), MR 88c:460 [MS] V.D. Milman and G. Schechtman, Asymtotic theory of finite dimensional normed saces, Lecture Notes in Math. 100, Sringer-Verlag 1986). MR 87m:46038 [PT] A. Pajor, N. Tomczak-Jaegermann, Subsaces of small codimension of finitedimensional Banach saces. Proc. Amer. Math. Soc ), no. 4, MR 87i:46040 [P] G. Pisier, The Volume of Convex Bodies and Banach Sace Geometry, Cambridge University Press 1989). MR 91d:5005 [R] S. Rolewicz, Metric linear saces. Monografie Matematyczne, Tom. 56. [Mathematical Monograhs, Vol. 56] PWN-Polish Scientific Publishers, Warsaw, 197. MR 55:10993 [S] V.N. Sudakov, Gaussian measures, Cauchy measures and ε-entroy. Soviet. Math. Dokl., ), Deartment of Mathematics, Tel Aviv University, Ramat Aviv, Israel address: alexandr@math.tau.ac.il Current address, A. Litvak: Deartment of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G G1 address: alexandr@math.ualberta.ca address: vitali@math.tau.ac.il Universite de Marne-la-Valle, Equie de Mathematiques, rue de la Butte Verte, 93166, Noisy-le-Grand Cedex, France address: ajor@math.univ-mlv.fr