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 � �� � 2 � f 2 d µ − |∇ f | 2 d µ. Var µ f = fd µ ≤ C µ Poincar´ e (spectral gap) constant = best value of C µ

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 E µ x i = 0 , I E µ ( x i x j ) = δ ij . I one has C K := C µ ≤ 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 C K ≤ c · C lin K , where Var µ ( f ) C lin K = sup K |∇ f | 2 d µ. � linear f

Other related conjectures and results 1) Hyperplane conjecture. ( Bourgain ). There exists universal c > 0 such that for any convex set K ⊂ R d 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: √ d ) 2 ) ≤ c I E µ (( | X | − for some universal constant c and every uniform distibution µ on an isotropic convex body K ⊂ R d . 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 ⊂ R d 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: √ d ) 2 ) ≤ c I E µ (( | X | − for some universal constant c and every uniform distibution µ on an isotropic convex body K ⊂ R d . 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 ⊂ R d 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: √ d ) 2 ) ≤ c I E µ (( | X | − for some universal constant c and every uniform distibution µ on an isotropic convex body K ⊂ R d . 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 ⊂ R d 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: √ d ) 2 ) ≤ c I E µ (( | X | − for some universal constant c and every uniform distibution µ on an isotropic convex body K ⊂ R d . 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 l p -balls B p = { x : | x 1 | p + | x 2 | p + · · · + | x d | p ≤ 1 } , 1 ≤ p ≤ ∞ . It is known that C B p ≤ c ( p ) 2 d p and B p 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 → ( ± x 1 , . . . , ± x d ) . K. Klartag [2009]: KLS conjecture holds for unconditional convex bodies up to the logarithmic factor C K ≤ C log d .

3. General bodies O. Gu´ edon, E. Milman [2011] and R. Eldan [2013] : 1 3 log d . C K ≤ Cd Best known result: Y.T. Lee, S. Vempala [2016] 1 4 . C K ≤ Cd

Main result Level set K E = { 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 1 d ( K E ) ≤ C C ≤ Vol and √ C K E ≤ C · C µ log( e + dC µ ) .

Removing logarithmic factor Theorem Let V i : R �→ R be convex functions, 1 ≤ i ≤ d. Assume that min( V i ) = 0 and every µ i = e − V i dx i is a probability measure witn barycenter at the origin. There exists a universal constant C > 1 and E ≤ d such that 1 1 d ( K E ) ≤ C C ≤ Vol and C lin � � C K E ≤ C log e + min( A 2 , d ) K E , where 1 A 2 = √ � ( α i ) � , d | V ′ � i ( y ) y − 1 | � � α i = inf t > 0: e dy ≤ e . t R

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: d p + p − � x ∈ R d ; � � � � K E := ( x i ) + + ( x i ) ≤ E , i i − i =1 satisfies 1 1 d ( K E ) ≤ C C ≤ Vol and C K E ≤ C log( e + min( P , d )) C lin K E for some universal C > 0.

Proof. Step 1. From µ to linearized measure on annulus Probability measure µ = e − V dx . Generalized ball K E = { V ≤ E } . Linearized probability measure µ K E = 1 � � e − E + n ( � x � KE − 1)) dx Z E Linearized probability measure on annulus 1 � � e − E + n ( � x � KE − 1)) µ K E ,ω = I 1 − ω ≤� x � KE ≤ 1 dx . Z E ,ω

Ratio of normalizing constants Note that � X K E � has Gamma distribution with respect to µ K E . In particular Z E = d ! e d d d e − E Vol ( K E ) , � 1 √ d d Z E ,ω e − dr r d − 1 dr ≥ c = d ω Z E ( d − 1)! 1 − ω 1 provided 0 ≤ ω ≤ d . √ Uniform estimate Corollary e d ω d µ K E ,ω ≤ √ d µ c d ω Z E

Ratio of normalizing constants Note that � X K E � has Gamma distribution with respect to µ K E . In particular Z E = d ! e d d d e − E Vol ( K E ) , � 1 √ d d Z E ,ω e − dr r d − 1 dr ≥ c = d ω Z E ( d − 1)! 1 − ω 1 provided 0 ≤ ω ≤ d . √ Uniform estimate Corollary e d ω d µ K E ,ω ≤ √ d µ c d ω Z E

Ratio of normalizing constants Note that � X K E � has Gamma distribution with respect to µ K E . In particular Z E = d ! e d d d e − E Vol ( K E ) , � 1 √ d d Z E ,ω e − dr r d − 1 dr ≥ c = d ω Z E ( d − 1)! 1 − ω 1 provided 0 ≤ ω ≤ d . √ Uniform estimate Corollary e d ω d µ K E ,ω ≤ √ d µ c d ω Z E

d µ KE ,ω e d ω We apply estimate ≤ √ d ω Z E and the following result d µ c Theorem [Barthe–Milman] Let ν 1 , ν 2 be probability measures. Assume that d ν 1 � L p ( ν 1 ) ≤ L for some p ∈ (1 , ∞ ] . Then setting q = p ∗ = � d ν 2 p p − 1 , we have: K 2 ( r ) ≤ 2 LK 1 / q ( r / 2) ∀ r > 0 , 1 where K 1 , K 2 are concentration functions of ν 1 , ν 2 . Thus we transfer concentration estimates from µ to µ E , K .

Step 2. From annulus to cone measure Note that the mapping x T ( x ) = � x � K E pushes forward the annulus measure µ K E ,ω onto the cone measure 1 � x , ν ∂ K E � · H d − 1 | ∂ K E . σ = Vol ( K E ) d To transfer concentration from the annulus measure to the cone measure we apply

Lemma [E. Milman] For every 1 -Lipshitz function f � � | f − med µ 2 f | d µ 2 ≤ | f − med µ 1 f | d µ 1 + W 1 ( µ 1 , µ 2 ) , where W 1 is the Kantorovich (Wasserstein) distance. and Lemma W 1 ( µ K E ,ω , σ ) ≤ d + 1 � ω � x � K E d λ K E , d where λ K E is the Lebesgue probability measure on K E .

Lemma [E. Milman] For every 1 -Lipshitz function f � � | f − med µ 2 f | d µ 2 ≤ | f − med µ 1 f | d µ 1 + W 1 ( µ 1 , µ 2 ) , where W 1 is the Kantorovich (Wasserstein) distance. and Lemma W 1 ( µ K E ,ω , σ ) ≤ d + 1 � ω � x � K E d λ K E , d where λ K E is the Lebesgue probability measure on K E .

Step 3. From cone measure to uniform measure Hardy-type inequality. For any convex Ω � � | f − med σ Ω | d σ Ω + 1 � | f − med λ Ω | d λ Ω ≤ |∇ f || x | d λ Ω , d Ω ∂ Ω where σ Ω is the corresponding cone measure. Proof: � ∂ Ω � x , ν Ω � gd H d − 1 � � � d gdx = Ω div ( x ) gdx = − Ω � x , ∇ g � dx + Ω ∂ Ω � x , ν Ω � gd H d − 1 . � � ≤ Ω | x ||∇ g | dx + Setting g = | f − med σ Ω | we complete the proof.

Last step Collecting everything together we get that for every 1-Lipschitz function f � | f − med λ Ω | d λ Ω ≤ C ( ω, d , Z E , C µ ) . Ω 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 d d e − d E ≥ 0 : e − E Vol ( K E ) ≥ 1 � � Level ( V ) = . e d ! Then Level ( V ) = [ E min , E max ] is a non-empty interval such that 1. E min ≤ d 2. √ 1 ≤ E min − E max ≤ e 2 π d (1 + o (1)) 3. 1 1 d ( E min ) ≤ Vol d ( E max ) ≤ ec ( d )(1 + o (1)) , c ( d ) ≤ Vol where c ( d ) → 1 as d → ∞ .