On KLS conjecture for certain classes of convex sets Alexander - - PowerPoint PPT Presentation

On KLS conjecture for certain classes of convex sets

Alexander Kolesnikov joint work with Emanuel Milman (Haifa)

Higher School of Economics, Moscow

B¸ edlewo, 2017

Poincar´ e inequality

We say that a probability measure µ satisfies the Poincar´ e inequality if Varµf =

• f 2dµ −
• fdµ

2 ≤ Cµ

• |∇f |2dµ.

Poincar´ e (spectral gap) constant = best value of Cµ

Motivating problem: KLS conjecture

Conjecture

Kannan, Lovasz, Simonovits (1995). There exists a universal number c such that for every uniform distribution µ on a convex body K satisfying I Eµxi = 0, I Eµ(xixj) = δij.

• ne has

CK := Cµ ≤ c. We call such bodies isotropic.

Equivalently:

there exists a universal number c such that for every uniform distribution µ on a convex body K CK ≤ c · C lin

K ,

where C lin

K =

sup linear f Varµ(f )

• K |∇f |2dµ.

Other related conjectures and results

1) Hyperplane conjecture. (Bourgain). There exists universal c > 0 such that for any convex set K ⊂ Rd of Vol(K) = 1 there exists a hyperplane L such that Vol(K ∩ L) > c. 2) Thin-shell conjecture. This is KLS conjecture for concrete function f = |X|. Can be put in the following way: I Eµ((|X| − √ d)2) ≤ c for some universal constant c and every uniform distibution µ

• n an isotropic convex body K ⊂ Rd.

3) Central limit theorem for convex bodies. Estimates of the thin-shell type = ⇒ central limit theorem for convex bodies (initiated by Sudakov (70’s), recent result: Klartag, 2006).

Other related conjectures and results

1) Hyperplane conjecture. (Bourgain). There exists universal c > 0 such that for any convex set K ⊂ Rd of Vol(K) = 1 there exists a hyperplane L such that Vol(K ∩ L) > c. 2) Thin-shell conjecture. This is KLS conjecture for concrete function f = |X|. Can be put in the following way: I Eµ((|X| − √ d)2) ≤ c for some universal constant c and every uniform distibution µ

• n an isotropic convex body K ⊂ Rd.

3) Central limit theorem for convex bodies. Estimates of the thin-shell type = ⇒ central limit theorem for convex bodies (initiated by Sudakov (70’s), recent result: Klartag, 2006).

Other related conjectures and results

1) Hyperplane conjecture. (Bourgain). There exists universal c > 0 such that for any convex set K ⊂ Rd of Vol(K) = 1 there exists a hyperplane L such that Vol(K ∩ L) > c. 2) Thin-shell conjecture. This is KLS conjecture for concrete function f = |X|. Can be put in the following way: I Eµ((|X| − √ d)2) ≤ c for some universal constant c and every uniform distibution µ

• n an isotropic convex body K ⊂ Rd.

3) Central limit theorem for convex bodies. Estimates of the thin-shell type = ⇒ central limit theorem for convex bodies (initiated by Sudakov (70’s), recent result: Klartag, 2006).

Other related conjectures and results

1) Hyperplane conjecture. (Bourgain). There exists universal c > 0 such that for any convex set K ⊂ Rd of Vol(K) = 1 there exists a hyperplane L such that Vol(K ∩ L) > c. 2) Thin-shell conjecture. This is KLS conjecture for concrete function f = |X|. Can be put in the following way: I Eµ((|X| − √ d)2) ≤ c for some universal constant c and every uniform distibution µ

• n an isotropic convex body K ⊂ Rd.

3) Central limit theorem for convex bodies. Estimates of the thin-shell type = ⇒ central limit theorem for convex bodies (initiated by Sudakov (70’s), recent result: Klartag, 2006).

What is known?

• 1. Special bodies

lp-balls Bp = {x : |x1|p + |x2|p + · · · + |xd|p ≤ 1}, 1 ≤ p ≤ ∞. It is known that CBp ≤ c(p) d

2 p

and Bp satisfies the KLS conjecture. Case 1 ≤ p ≤ 2: S. Sodin [2008] Case 2 ≤ p ≤ ∞: R. Lata la, J. O. Wojtaszczyk [2008] Simplex: F. Barthe, P. Wolff [2009].

• 2. Unconditional convex bodies

Unconditional bodies are bodies which are invariant with respect to the mappings x → (±x1, . . . , ±xd).

• K. Klartag [2009]: KLS conjecture holds for unconditional convex

bodies up to the logarithmic factor CK ≤ C log d.

• 3. General bodies
• O. Gu´

edon, E. Milman [2011] and R. Eldan [2013] : CK ≤ Cd

1 3 log d.

Best known result: Y.T. Lee, S. Vempala [2016] CK ≤ Cd

1 4 .

Main result

Level set KE = {V ≤ E}.

Theorem

Let µ = e−V dx be a log-concave probability measure with min(V ) = 0. There exists a universal constant C > 1 and E ≤ d such that 1 C ≤ Vol

1 d (KE) ≤ C

and CKE ≤ C · Cµ log(e + √ dCµ).

Removing logarithmic factor

Theorem

Let Vi : R → R be convex functions, 1 ≤ i ≤ d. Assume that min(Vi) = 0 and every µi = e−Vidxi is a probability measure witn barycenter at the origin. There exists a universal constant C > 1 and E ≤ d such that 1 C ≤ Vol

1 d (KE) ≤ C

and CKE ≤ C log

• e + min(A2, d)
• C lin

KE ,

where A2 = 1 √ d (αi), αi = inf

• t > 0:
• R

e

|V ′ i (y)y−1| t

dy ≤ e

• .

Example

Let p±

i ∈ [1, P], i = 1, . . . , d, for a fixed arbitrary constant P ≥ 1.

Then there exists E ≤ d so that the generalized Orlicz ball: KE :=

• x ∈ Rd ;

d

• i=1
• (xi)

p+

i

+ + (xi) p−

i

−

• ≤ E
• ,

satisfies 1 C ≤ Vol

1 d (KE) ≤ C

and CKE ≤ C log(e + min(P, d))C lin

KE

for some universal C > 0.

• Proof. Step 1. From µ to linearized measure on annulus

Probability measure µ = e−V dx. Generalized ball KE = {V ≤ E}. Linearized probability measure µKE = 1 ZE e−

• E+n(xKE −1))
• dx

Linearized probability measure on annulus µKE,ω = 1 ZE,ω e−

• E+n(xKE −1))
• I1−ω≤xKE ≤1dx.

Ratio of normalizing constants

Note that XKE has Gamma distribution with respect to µKE . In particular ZE = d!ed dd e−EVol(KE), ZE,ω ZE = dd (d − 1)! 1

1−ω

e−drrd−1dr ≥ c √ dω provided 0 ≤ ω ≤

1 √ d .

Uniform estimate Corollary

dµKE,ω dµ ≤ edω c √ dωZE

Ratio of normalizing constants

Note that XKE has Gamma distribution with respect to µKE . In particular ZE = d!ed dd e−EVol(KE), ZE,ω ZE = dd (d − 1)! 1

1−ω

e−drrd−1dr ≥ c √ dω provided 0 ≤ ω ≤

1 √ d .

Uniform estimate Corollary

dµKE,ω dµ ≤ edω c √ dωZE

Ratio of normalizing constants

Note that XKE has Gamma distribution with respect to µKE . In particular ZE = d!ed dd e−EVol(KE), ZE,ω ZE = dd (d − 1)! 1

1−ω

e−drrd−1dr ≥ c √ dω provided 0 ≤ ω ≤

1 √ d .

Uniform estimate Corollary

dµKE,ω dµ ≤ edω c √ dωZE

We apply estimate

dµKE,ω dµ

≤

edω c √ dωZE and the following result

Theorem

[Barthe–Milman] Let ν1, ν2 be probability measures. Assume that dν2

dν1 Lp(ν1) ≤ L for some p ∈ (1, ∞]. Then setting q = p∗ = p p−1,

we have: K2(r) ≤ 2LK 1/q

1

(r/2) ∀r > 0, where K1, K2 are concentration functions of ν1, ν2. Thus we transfer concentration estimates from µ to µE,K.

Step 2. From annulus to cone measure

Note that the mapping T(x) = x xKE pushes forward the annulus measure µKE,ω onto the cone measure σ = 1 Vol(KE) x, ν∂KE d · Hd−1|∂KE . To transfer concentration from the annulus measure to the cone measure we apply

Lemma

[E. Milman] For every 1-Lipshitz function f

• |f − medµ2f |dµ2 ≤
• |f − medµ1f |dµ1 + W1(µ1, µ2),

where W1 is the Kantorovich (Wasserstein) distance. and

Lemma

W1(µKE,ω, σ) ≤ d + 1 d ω

• xKE dλKE ,

where λKE is the Lebesgue probability measure on KE.

Lemma

[E. Milman] For every 1-Lipshitz function f

• |f − medµ2f |dµ2 ≤
• |f − medµ1f |dµ1 + W1(µ1, µ2),

where W1 is the Kantorovich (Wasserstein) distance. and

Lemma

W1(µKE,ω, σ) ≤ d + 1 d ω

• xKE dλKE ,

where λKE is the Lebesgue probability measure on KE.

Step 3. From cone measure to uniform measure

Hardy-type inequality. For any convex Ω

• Ω

|f − medλΩ|dλΩ ≤

• ∂Ω

|f − medσΩ|dσΩ + 1 d

• |∇f ||x|dλΩ,

where σΩ is the corresponding cone measure. Proof: d

• Ω

gdx =

• Ω div(x)gdx = −
• Ωx, ∇gdx +
• ∂Ωx, νΩgdHd−1

≤

• Ω |x||∇g|dx +
• ∂Ωx, νΩgdHd−1.

Setting g = |f − medσΩ| we complete the proof.

Last step

Collecting everything together we get that for every 1-Lipschitz function f

• Ω

|f − medλΩ|dλΩ ≤ C(ω, d, ZE, Cµ). Optimize parameters: choosing ω = 1

d and applying lemma below

we get the result.

Lemma

Let µ = e−V dx be a log-concave probability measure with min(V ) = 0. Set Level(V ) =

• E ≥ 0 : e−EVol(KE) ≥ 1

e dde−d d!

• .

Then Level(V ) = [Emin, Emax] is a non-empty interval such that 1. Emin ≤ d 2. 1 ≤ Emin − Emax ≤ e √ 2πd(1 + o(1)) 3. c(d) ≤ Vol

1 d (Emin) ≤ Vol 1 d (Emax) ≤ ec(d)(1 + o(1)),

where c(d) → 1 as d → ∞.

Lemma

Let µ = e−V dx be a log-concave probability measure with min(V ) = 0. Set Level(V ) =

• E ≥ 0 : e−EVol(KE) ≥ 1

e dde−d d!

• .

Then Level(V ) = [Emin, Emax] is a non-empty interval such that 1. Emin ≤ d 2. 1 ≤ Emin − Emax ≤ e √ 2πd(1 + o(1)) 3. c(d) ≤ Vol

1 d (Emin) ≤ Vol 1 d (Emax) ≤ ec(d)(1 + o(1)),

where c(d) → 1 as d → ∞.

Lemma

Let µ = e−V dx be a log-concave probability measure with min(V ) = 0. Set Level(V ) =

• E ≥ 0 : e−EVol(KE) ≥ 1

e dde−d d!

• .

Then Level(V ) = [Emin, Emax] is a non-empty interval such that 1. Emin ≤ d 2. 1 ≤ Emin − Emax ≤ e √ 2πd(1 + o(1)) 3. c(d) ≤ Vol

1 d (Emin) ≤ Vol 1 d (Emax) ≤ ec(d)(1 + o(1)),

where c(d) → 1 as d → ∞.

Lemma

Let µ = e−V dx be a log-concave probability measure with min(V ) = 0. Set Level(V ) =

• E ≥ 0 : e−EVol(KE) ≥ 1

e dde−d d!

• .

Then Level(V ) = [Emin, Emax] is a non-empty interval such that 1. Emin ≤ d 2. 1 ≤ Emin − Emax ≤ e √ 2πd(1 + o(1)) 3. c(d) ≤ Vol

1 d (Emin) ≤ Vol 1 d (Emax) ≤ ec(d)(1 + o(1)),

where c(d) → 1 as d → ∞.