Convex inequalities, isoperimetry and spectral gap III

Transcription

1 Convex inequalities, isoperimetry and spectral gap III Jesús Bastero (Universidad de Zaragoza) CIDAMA Antequera, September 11, 2014

2 Part III. K-L-S spectral gap conjecture KLS estimate, through Milman's theorem Gaussian case Other related conjectures

3 Kannan-Lovász-Simonovits problem Given µ a log-concave probability on R n, estimate the biggest constants µ + (A) Cµ(A) borelian, µ(a) 1 2

4 Kannan-Lovász-Simonovits problem Given µ a log-concave probability on R n, estimate the biggest constants µ + (A) Cµ(A) borelian, µ(a) 1 2 We know λ 2 E µ f E µ f 2 E µ f 2 C E µ f E µ f 2 1 locally Lipschitz f 1-Lipschitz f C 2 λ 2 C 2

5 Kannan-Lovász-Simonovits conjecture KLS conjecture The biggest constant is attained for ane functions, up to an absolute constant.

6 Kannan-Lovász-Simonovits conjecture KLS conjecture The biggest constant is attained for ane functions, up to an absolute constant. f ane, f (x) = t + a, x, t R, a R n f ane 1-Lipschitz, f (x) = t + θ, x, t R, θ S n 1 E µ f = t + θ, E µ x Then, Compute E µ f E µ f 2 = E µ θ, x E µ x 2 λ 2 µ := sup θ S n 1 E µ θ, x E µ x 2 (λ µ := highest eigenvalue of the covariance matrix of µ)

7 Kannan-Lovász-Simonovits conjecture Let µ a log-concave probability in R n with barycenter E µ x. Let λ 2 µ = then Poincaré inequality is sup θ S n 1 E µ x E µ x, θ 2 E µ f E µ f 2 Cλ 2 µ E µ f 2

8 Kannan-Lovász-Simonovits conjecture Let µ a log-concave probability in R n with barycenter E µ x. Let λ 2 µ = then Poincaré inequality is sup θ S n 1 E µ x E µ x, θ 2 E µ f E µ f 2 Cλ 2 µ E µ f 2 C > 0 is an absolute constant, independent of µ and even of the dimension, λ 2 µ is also the highest eigenvalue for the covariance matrix for µ, say A = E µ (x E µ x) (x E µ x) = ( E µ (x i x j E µ x i E µ x j ) 2) 1 i,j n

9 KLS estimate Given µ (log-concave) and f Lipschitz E µ f E µ f 2 CE µ x E µ x 2 E µ f 2 C > 0 is an absolute constant.

10 KLS estimate Given µ (log-concave) and f Lipschitz C > 0 is an absolute constant. Proof. E µ f E µ f 2 CE µ x E µ x 2 E µ f 2 E µ f E µ f 2 E µ ( f (x) f (Eµ x) + f (E µ x) E µ f ) 2 = E µ ( f (x) f (Eµ x) + E µ (f (E µ x) f ) ) 2 (by Minkowski's inequality) 4 E µ f (x) f (E µ x) 2 4E µ x E µ x 2 f 2 ( C E µ x E µ x 2 E µ f 2 C n }{{} λ2 µ E µ f 2) E µ x E µ x 2 = n i=1 E µ(x i E µ x i ) 2

11 Previous results Payne-Weinberger (1960) If µ is the normalized measure in a convex body K E µ f E µ f 2 4 π 2 diam(k)2 E µ f 2

12 Previous results Payne-Weinberger (1960) If µ is the normalized measure in a convex body K E µ f E µ f 2 4 π 2 diam(k)2 E µ f 2 If B Euclidean ball and µ its normalized measure (the right estimate) E µ f E µ f 2 C n E µ f 2

13 Examples of product spaces Talagrand (1991) Let dµ(x) = 1 2 n e n i=1 xi dx. Then E µ f E µ f 2 C E µ f 2

14 Examples of product spaces Talagrand (1991) Let dµ(x) = 1 2 n e n i=1 xi dx. Then E µ f E µ f 2 C E µ f 2 Gaussian case Let dµ(x) = (2π) n/2 e x 2 /2. Then Var µ f = E µ f E µ f 2 E µ f 2

15 Proof of Gaussian case It is easy to see that λ µ = 1 Let now u D(R n ), test functions. Consider the associated Laplace Beltrami operator L Lu(x) = u(x) x, u(x)

16 Proof of Gaussian case It is easy to see that λ µ = 1 Let now u D(R n ), test functions. Consider the associated Laplace Beltrami operator L We know: Lu(x) = u(x) x, u(x) {Lu; u D} is dense in {f L 2 (R n, dµ) : E µ f = 0} Then, inf u D { Eµ (Lu f ) 2} = 0

17 Proof of Gaussian case It is easy to see that λ µ = 1 Let now u D(R n ), test functions. Consider the associated Laplace Beltrami operator L We know: Lu(x) = u(x) x, u(x) {Lu; u D} is dense in {f L 2 (R n, dµ) : E µ f = 0} Then, inf u D { Eµ (Lu f ) 2} = 0 and integrating by parts E µ f Lu = E µ f, u (Green formula) E µ (Lu) 2 = E µ u, u + E µ i,j ( ij u(x)) 2 E µ u 2

18 Assume E µ f = 0. Since Var µ f = E f µ Eµ f 2 = E µ f 2

19 Assume E µ f = 0. Since Var µ f = E f µ Eµ f 2 = E µ f 2 E µ f 2 E µ (Lu f ) 2 = 2E µ f Lu E µ (Lu) 2 2E µ f, u E µ u 2 E µ f 2

20 Assume E µ f = 0. Since Var µ f = E f µ Eµ f 2 = E µ f 2 E µ f 2 E µ (Lu f ) 2 = 2E µ f Lu E µ (Lu) 2 2E µ f, u E µ u 2 E µ f 2 Taking the inmum in u we obtain the result. Var µ f λ 2 µ E µ f 2

21 Known results The normalized measure on p-balls, 1 p (Sodin 2008, Šatala&Oleskievicz 2008) The simplex (Barthe and Wol, 2009) Some revolution bodies (Bobkov, 2003, Hue) Unconditional bodies (Klartag, 2009)with a log n constant, i.e. Var µ f = E f µ Eµ f 2 C log n λ 2 µ E µ f 2 K is unconditional: (x 1,..., x n ) K i ( x 1,..., x n ) K

22 Better known estimate Guedon-Milman (2011) + Eldan (2013) Poincaré inequality is true in the following way Var µ f = E f µ Eµ f 2 C n 2/3 (log n) 2 λ 2 µ E µ f 2 for any locally Lipschitz integrable function f.

23 Geometric connections: Three conjectures The slicing problem ( 1986, Bourgain) Thin shell width conjecture (2003, Bobkov-Koldobsky, Antilla-Ball-Perissinaki) Kannan-Lovász-Simonovits spectral gap conjecture (1995)

24 Geometric connections: Three conjectures The slicing problem ( 1986, Bourgain) (Eldan-Klartag 2010) Thin shell width conjecture (2003, Bobkov-Koldobsky, Antilla-Ball-Perissinaki) Kannan-Lovász-Simonovits spectral gap conjecture (1995)

25 Geometric connections: Three conjectures The slicing problem ( 1986, Bourgain) (Eldan-Klartag 2010) Thin shell width conjecture (2003, Bobkov-Koldobsky, Antilla-Ball-Perissinaki) (log n) 2 (Eldan 2013) Kannan-Lovász-Simonovits spectral gap conjecture (1995)

26 Geometric connections: Three conjectures The slicing problem ( 1986, Bourgain) (Eldan-Klartag 2010) Thin shell width conjecture (2003, Bobkov-Koldobsky, Antilla-Ball-Perissinaki) (log n) 2 (Eldan 2013) Kannan-Lovász-Simonovits spectral gap conjecture (1995) The slicing problem (Ball-Nguyen 2013)

27 The slicing problem Conjecture (Bourgain, 1986) There exists an absolute constant C > 0 such that every convex body K in R n with volume 1 has, at least, one (n 1)-dimensional section such that K H n 1 C (Optimistic estimate C = 1 e!)

28 Slicing problem is true in Unconditional convex bodies Zonoids Random polytopes Polytopes in which the number of vertices is proportional to de dimension, i.e., for instance, N/n 2 The unit balls of nite dimensional Schatten classes,for 1 p (n 1)-orthogonal projection of the classes above and more

29 Known general estimates C Bourgain (1986), K H n 1 n 1/4 log n Klartag (2006), K H n 1 C n 1/4

30 Thin shell width conjecture (2003, Bobkov-Koldobsky, Antilla-Ball-Perissinaki) Conjecture There exists an absolute constant C > 0 such that for every log-concave probability µ in R n Remark: σ µ = E µ x E µ x 2 Cλ µ. It is equivalent to KLS conjecture to be true only for the function x or x 2

31 The name is due to the following fact If the thin shell width conjecture were true, we would have a stronger concentration of the mass around the mean for log-concave probabilities µ { x Eµ x > teµ x } 2 exp C t 1 (E µ x ) 1 2 2, t > 0 λ 1 2 µ

32 Thin shell width conjecture is true for The uniform probability on Finite dimensional p-balls, 1 p Finite dimensional Orlicz -balls Revolution bodies (n 1)-dimensional orthogonal projections of the crosspolytope (1-ball) (n 1)-dimensional orthogonal projections of the cube and even all their linear deformations Remark: This conjecture is not linear invariant, but in a random sense, more than half of linear deformations of the classe above also satisfy this conjecture

33 Best known estimate Guedon-Milman, 2010 There exists an absolute constant C > 0 such that for every log-concave probability µ in R n σ µ C n 1/3 λ µ.

34 Our last contribution in the thin shell width conjecture, joint work with D. Alonso (2013 +?) (n 1)-dimensional orthogonal projections of the crosspolytope (1-ball) satisfy this conjecture (n 1)-dimensional orthogonal projections of the cube and even all their linear deformation satisfy this conjecture (n 1)-dimensional orthogonal projections of the p-balls satisfy this conjecture whe the vectors on which the projection is taken are sparse. If µ satises the t.s.w. conjecture then ν = µ T also satises the t.s.w. conjecture at least for half of T 's (T linear mapp) in a probabilistis meaning and 'at random' if Schatten norm of T c 4 satises T HS = o(n 1 4 ) T c 4

35 References D. Alonso-Gutiérrez, J. Bastero, The variance conjecture on some polytopes. In Asymptotic Geometric Analysis. Proceedings of the Fall 2010, Fields InstituteTematic Program. Springer (2013), 120. D. Alonso-Gutiérrez, J. Bastero, Approaching the Kannan-Lovász-Simonovits and variance conjectures, preprint S. Bobkov, C. Houdré, Some connections between isoperimetric and Sobolev-type inequalities. Mem. Amer. Math. Soc., 129(616), S. Brazitikos, A. Giannopoulos, P. Valettas and BH. Vritsiou, Geometry of Isotropic Log-Concave measures. to appear in AMS-Mathematical Surveys and Monographs. J. Cheeger, A lower bound for the smallest eigenvalue of the Laplacian: In Problems in Analysis (Papers dedicated to Salomon Bochner, 1969) Princeton Univ. Press. Princeton (1970) R. Eldan, Thin shell implies spectral gap up to polylog via stochastic localization scheme. Geom. Funct. Anal. Vol.23, (2013),

36 O. Guedon, E. Milman, Interpolating thin-shell and sharp large-deviation estimates for isotropic log-concave measures, Preprint (2011)Geom. Funct.l Anal., 21(5), (2011), R. Kannan, L. Lovász, M. Simonovits Isoperimetric problems for convex bodies and a localization lemma. Discrete Comput. Geom. 13, no. 3-4, (1995), B. Klartag, A Berry-Esseen type inequality for convex bodies with an unconditional basis. Probab. Theory Related Fields 145, no. 1-2, (2009), 133. R. Latala and J. O. Wojtaszczyk On the inmum convolution inequality. Studia Math., 189(2), (2008), M. Ledoux The concentration of measure phenomenon. Math. Surveys and Monographs A.M.S. 89, (2001). E. Milman, On the role of convexity in isoperimetry, spectral gap and concentration. Invent. Math. 177 (2009), no. 1, 143.

37 V.D. Milman, A. Pajor, Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space GAFA Seminar 87-89, Springer Lecture Notes in Math, no (1989), pp V.D. Milman, G. Schechtman, Asymptotic Theory of Finite Dimensional Normed Spaces Springer Lecture Notes in Math, no (1986) G. Paouris, Concentration of mass on convex bodies, Geom. and Funct. Anal. (GAFA) 16, (2006) S. Sodin, An isoperimetric inequality on the l p balls. Ann. Inst. H. Poincar é Probab. Statist. 44, no. 2, (2008), M. Talagrand, A new isoperimetric inequality and the concentration of measure phenomenon. En Geometric aspects of functional analysis, volume 1469 of Lecture Notes in Math., page Springer, Berlin, 1991.