Mean-width and mean-norm of isotropic convex bodies Aleksander Pe� lczy´ nski Memorial Conference Apostolos Giannopoulos July 16, 2014 M and M ∗ estimates (Bedlewo 2014) July 16, 2014 1 / 36
M and M ∗ We assume that K is a centrally symmetric convex body of volume 1 in R n : K = { x ∈ R n : � x � � 1 } . The mean-norm of K is defined by � M ( K ) = S n − 1 � x � d σ ( x ) . M and M ∗ estimates (Bedlewo 2014) July 16, 2014 2 / 36
M and M ∗ We assume that K is a centrally symmetric convex body of volume 1 in R n : K = { x ∈ R n : � x � � 1 } . The mean-norm of K is defined by � M ( K ) = S n − 1 � x � d σ ( x ) . The support function of K is h K ( x ) = � x � ∗ = max {� x , y � : y ∈ K } , and the mean-width of K is � M ∗ ( K ) = w ( K ) = S n − 1 h K ( x ) d σ ( x ) . M and M ∗ estimates (Bedlewo 2014) July 16, 2014 2 / 36
Lower bounds Using integration in polar coordinates and H¨ older’s inequality we get � 1 / n � | B n 2 | � c 1 √ n . M ( K ) � | K | From Urysohn’s inequality, � | K | � 1 / n √ n . M ∗ ( K ) � vrad ( K ) := � c 2 | B n 2 | M and M ∗ estimates (Bedlewo 2014) July 16, 2014 3 / 36
Lower bounds Using integration in polar coordinates and H¨ older’s inequality we get � 1 / n � | B n 2 | � c 1 √ n . M ( K ) � | K | From Urysohn’s inequality, � | K | � 1 / n √ n . M ∗ ( K ) � vrad ( K ) := � c 2 | B n 2 | These lower bounds for M and M ∗ are sharp: if D n = r n B n 2 has volume 1 then r n ≃ √ n and M ∗ ( D n ) = r n ≃ √ n . M ( D n ) = 1 1 ≃ √ n while r n M and M ∗ estimates (Bedlewo 2014) July 16, 2014 3 / 36
Upper bounds Theorem (Lewis, Figiel-Tomczak, Pisier) Every centrally symmetric convex body K in R n has a linear image ( a position ) ˜ K of volume 1 such that M ( ˜ K ) M ∗ ( ˜ K ) � c 1 log[ d ( X K , ℓ n 2 ) + 1] � c 2 log n . M and M ∗ estimates (Bedlewo 2014) July 16, 2014 4 / 36
Upper bounds Theorem (Lewis, Figiel-Tomczak, Pisier) Every centrally symmetric convex body K in R n has a linear image ( a position ) ˜ K of volume 1 such that M ( ˜ K ) M ∗ ( ˜ K ) � c 1 log[ d ( X K , ℓ n 2 ) + 1] � c 2 log n . For this position of K , using the previous lower bounds, we have K ) � c √ n log n . K ) � c log n M ( ˜ M ∗ ( ˜ √ n and Question: What can we say about the isotropic position? M and M ∗ estimates (Bedlewo 2014) July 16, 2014 4 / 36
Isotropic convex bodies Isotropic convex bodies A convex body K in R n is called isotropic if it has volume 1, it is centered, and there exists a constant L K > 0 such that � � x , θ � 2 dx = L 2 K K for every θ ∈ S n − 1 . M and M ∗ estimates (Bedlewo 2014) July 16, 2014 5 / 36
Isotropic convex bodies Isotropic convex bodies A convex body K in R n is called isotropic if it has volume 1, it is centered, and there exists a constant L K > 0 such that � � x , θ � 2 dx = L 2 K K for every θ ∈ S n − 1 . Hyperplane conjecture There exists an absolute constant C > 0 such that L K � C for every n and every isotropic convex body K in R n . √ n log n , Klartag: L K � c 4 √ n . Bourgain: L K � c 4 M and M ∗ estimates (Bedlewo 2014) July 16, 2014 5 / 36
Log-concave measures Log-concave measures A measure µ on R n is called log-concave if µ ( λ A + (1 − λ ) B ) � µ ( A ) λ µ ( B ) 1 − λ for any non-empty compact subsets A and B of R n and any λ ∈ (0 , 1). M and M ∗ estimates (Bedlewo 2014) July 16, 2014 6 / 36
Log-concave measures Log-concave measures A measure µ on R n is called log-concave if µ ( λ A + (1 − λ ) B ) � µ ( A ) λ µ ( B ) 1 − λ for any non-empty compact subsets A and B of R n and any λ ∈ (0 , 1). Isotropic log-concave measures We say that a log-concave probability measure µ is isotropic if bar ( µ ) = 0 and Cov ( µ ) is the identity matrix: � x i x j f µ ( x ) dx = δ ij . Then, the isotropic constant of µ is L µ = � f µ � 1 / n ∞ ≃ f µ (0) 1 / n . M and M ∗ estimates (Bedlewo 2014) July 16, 2014 6 / 36
Isotropic log-concave measures If K is a convex body in R n , then the Brunn-Minkowski inequality implies that 1 K is the density of a log-concave measure. K is isotropic if and only if the measure µ K with density L n K 1 1 LK K is isotropic. M and M ∗ estimates (Bedlewo 2014) July 16, 2014 7 / 36
Isotropic log-concave measures If K is a convex body in R n , then the Brunn-Minkowski inequality implies that 1 K is the density of a log-concave measure. K is isotropic if and only if the measure µ K with density L n K 1 1 LK K is isotropic. Marginal The marginal of µ with respect to F ∈ G n , k is defined by π F µ ( A ) := µ ( P − 1 F ( A )) = µ ( A + F ⊥ ) for all Borel subsets of F . The density of π F µ is the function � f π F µ ( x ) = x + F ⊥ f µ ( y ) dy , x ∈ F . If µ is centered, log-concave or isotropic, then π F µ is respectively also centered, log-concave or isotropic. M and M ∗ estimates (Bedlewo 2014) July 16, 2014 7 / 36
L q -centroid bodies L q -centroid bodies If µ is a probability measure on R n , the L q -centroid body Z q ( µ ), q � 1, is the symmetric convex body with support function � 1 / q �� |� x , y �| q d µ ( x ) h Z q ( µ ) ( y ) := ��· , y �� L q ( µ ) = . M and M ∗ estimates (Bedlewo 2014) July 16, 2014 8 / 36
L q -centroid bodies L q -centroid bodies If µ is a probability measure on R n , the L q -centroid body Z q ( µ ), q � 1, is the symmetric convex body with support function � 1 / q �� |� x , y �| q d µ ( x ) h Z q ( µ ) ( y ) := ��· , y �� L q ( µ ) = . µ is isotropic if and only if it is centered and Z 2 ( µ ) = B n 2 . From H¨ older’s inequality it follows that Z 2 ( µ ) ⊆ Z p ( µ ) ⊆ Z q ( µ ) for all 2 � p � q < ∞ . From Borell’s lemma, Z q ( µ ) ⊆ c q p Z p ( µ ) for all 2 � p < q . If µ is isotropic, then R ( Z q ( µ )) := max { h Z q ( µ ) ( θ ) : θ ∈ S n − 1 } � cq . M and M ∗ estimates (Bedlewo 2014) July 16, 2014 8 / 36
L q -centroid bodies L q -centroid bodies If K is a convex body of volume 1 in R n , the L q -centroid body Z q ( K ), q � 1, is the symmetric convex body with support function � 1 / q �� |� x , y �| q dx h Z q ( K ) ( y ) := . K K is isotropic if and only if it is centered and Z 2 ( K ) = L K B n 2 . If K is centrally symmetric then cK ⊆ Z q ( K ) ⊆ K for all q � n . If K is isotropic and if µ K is the isotropic measure with density L n K 1 1 LK K , then Z q ( K ) = L K Z q ( µ K ) . M and M ∗ estimates (Bedlewo 2014) July 16, 2014 9 / 36
The two questions Assume that K is centrally symmetric and isotropic in R n . Question 1 To give an upper bound for M ∗ ( K ). From the inclusion K ⊆ ( n + 1) L K B n 2 , one has the obvious bound M ∗ ( K ) � ( n + 1) L K . Until recently, it was known that M ∗ ( K ) � cn 3 / 4 L K . Several approaches: Hartzoulaki, Pivovarov, “ Z q -bound”. M and M ∗ estimates (Bedlewo 2014) July 16, 2014 10 / 36
The two questions Assume that K is centrally symmetric and isotropic in R n . Question 1 To give an upper bound for M ∗ ( K ). From the inclusion K ⊆ ( n + 1) L K B n 2 , one has the obvious bound M ∗ ( K ) � ( n + 1) L K . Until recently, it was known that M ∗ ( K ) � cn 3 / 4 L K . Several approaches: Hartzoulaki, Pivovarov, “ Z q -bound”. Question 2 To give an upper bound for M ( K ). From the inclusion K ⊇ L K B n 2 , one has the obvious bound M ( K ) � 1 / L K . Until recently, there was no lower bound depending on n . M and M ∗ estimates (Bedlewo 2014) July 16, 2014 10 / 36
A first idea: Dudley’s entropy estimate Entropy numbers The covering number N ( K , T ) of K by T is the minimal number of translates of T whose union covers K . For any k � 1 we set e k ( K , T ) := inf { s > 0 : N ( K , sT ) � 2 k } . The k -th entropy number of K is e k ( K ) := e k ( K , B n 2 ). M and M ∗ estimates (Bedlewo 2014) July 16, 2014 11 / 36
A first idea: Dudley’s entropy estimate Entropy numbers The covering number N ( K , T ) of K by T is the minimal number of translates of T whose union covers K . For any k � 1 we set e k ( K , T ) := inf { s > 0 : N ( K , sT ) � 2 k } . The k -th entropy number of K is e k ( K ) := e k ( K , B n 2 ). Dudley’s bound If K is a centrally symmetric convex body in R n then √ nM ∗ ( K ) � c 1 1 � e k ( K , B n √ 2 ) . k k � 1 M and M ∗ estimates (Bedlewo 2014) July 16, 2014 11 / 36
A first idea: Dudley’s entropy estimate Covering numbers If K is an isotropic convex body in R n then n 3 / 2 L K log N ( K , sB n 2 ) � C 1 s for all s > 0. Therefore, √ nL K n e k ( K , B n 2 ) = inf { s > 0 : N ( K , sB n 2 ) � 2 k } � C 2 k . M and M ∗ estimates (Bedlewo 2014) July 16, 2014 12 / 36
A first idea: Dudley’s entropy estimate Covering numbers If K is an isotropic convex body in R n then n 3 / 2 L K log N ( K , sB n 2 ) � C 1 s for all s > 0. Therefore, √ nL K n e k ( K , B n 2 ) = inf { s > 0 : N ( K , sB n 2 ) � 2 k } � C 2 k . Then, we combine this with Dudley’s bound 1 M ∗ ( K ) � c 1 � e k ( K , B n √ 2 ) k k � 1 to get M ∗ ( K ) � Cn 3 / 4 L K . M and M ∗ estimates (Bedlewo 2014) July 16, 2014 12 / 36
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