Bounds for the volume ratio of convex bodies DANIEL GALICER (Joint - PowerPoint PPT Presentation

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work Remarks If L = B n 2 we recover the standard notion of volume ratio. That is, vr( K , B n 2 ) = vr( K ) . The volume ratio is affinely invariant: vr( K , L ) = vr( T ( K ) , S ( L )) , for every affine transformation T , S .

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work Remarks If L = B n 2 we recover the standard notion of volume ratio. That is, vr( K , B n 2 ) = vr( K ) . The volume ratio is affinely invariant: vr( K , L ) = vr( T ( K ) , S ( L )) , for every affine transformation T , S . vr( K , L ) ≤ vr( K , Z )vr( Z , L ).

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work Remarks If L = B n 2 we recover the standard notion of volume ratio. That is, vr( K , B n 2 ) = vr( K ) . The volume ratio is affinely invariant: vr( K , L ) = vr( T ( K ) , S ( L )) , for every affine transformation T , S . vr( K , L ) ≤ vr( K , Z )vr( Z , L ). vr( K , L ) ≈ vr( L ◦ , K ◦ ) .

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work General upper bounds for the volume ratio Giannopoulos-Hartzoulaki (2002) For every pair of convex bodies K , L ⊂ R n vr( K , L ) ≤ C √ n log( n ) . [GH02] A Giannopoulos, M Hartzoulaki On the volume ratio of two convex bodies. Bulletin of the London Mathematical Society 34.6 (2002): 703-707.

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work Lower bounds Khrabrov (2001) Given a convex body K there is Z such that n � vr( K , Z ) ≥ C log log( n ) . [Khr01] Alexander Igorevich Khrabrov. Generalized volume ratios and the Banach–Mazur distance. Mathematical Notes , 2001.

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work The largest volume ratio Given a convex body K ⊂ R n we define its largest volume ratio as lvr( K ) := sup L ⊂ R n vr ( K , L ) . Remarks For every convex body K we have: log log( n ) � lvr( K ) � √ n log( n ) . � n

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work The largest volume ratio Given a convex body K ⊂ R n we define its largest volume ratio as lvr( K ) := sup L ⊂ R n vr ( K , L ) . Remarks For every convex body K we have: log log( n ) � lvr( K ) � √ n log( n ) . � n For many bodies K ⊂ R n , lvr( K ) ≈ √ n .

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work Question: Can we improve the general bounds?

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work Question: Can we improve the general bounds? Question: Can we show sharp asymptotic estimates for certain classes of convex bodies?

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work Lower estimates...

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work Recall... For every convex body K , the best known general bounds for the largest volume ratio are log log( n ) � lvr( K ) � √ n log( n ) � n

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work Recall... For every convex body K , the best known general bounds for the largest volume ratio are log log( n ) � lvr( K ) � √ n log( n ) � n G., Merzbacher, Pinasco, 2019+ For every convex body K we have: √ n � lvr( K )

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work Reduction to the symmetric case Rogers-Shephard Inequality Given a convex body K , 1 1 n ≤ 4 vol ( K ) n . vol ( K − K )

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work Reduction to the symmetric case Rogers-Shephard Inequality Given a convex body K , 1 1 n ≤ 4 vol ( K ) n . vol ( K − K ) A body which contains the origin can be approximated by outside by a symmetric body with essentially the same volume. vr( K − K , K ) ≤ 4 . Then, vr( K − K , Z ) ≤ vr( K − K , K )vr( K , Z ) ≤ 4vr( K , Z ) .

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work Assumption Since there is always a reduction (considering either K − K or K ∩ − K depending on the case) we will assume for simplicity that all bodies involved are centrally symmetric.

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work If K and Z are centrally symmetric 1 vr( K , Z ) = vol ( K ) n · inf {� T : X Z → X K � : det( T ) = 1 } 1 vol ( Z ) n

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work If K and Z are centrally symmetric 1 vr( K , Z ) = vol ( K ) n · inf {� T : X Z → X K � : det( T ) = 1 } 1 vol ( Z ) n Idea: Given a fixed centrally symmetric body K ⊂ R n , find Z such that for every T ∈ SL n ( R ) the norm � T : X Z → X K � is big, 1 n is small. the measure vol ( Z ) But how???

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work How? The probabilistic method... ... is a nonconstructive method for proving the existence of a prescribed kind of mathematical object.

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work How? The probabilistic method... ... is a nonconstructive method for proving the existence of a prescribed kind of mathematical object. It works by showing that if one randomly chooses objects from a specified class, the probability that the result is of the prescribed kind is strictly greater than zero.

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work Random polytopes on the sphere (Gluskin’s polytopes) Given m ∈ N , we consider the set A m := { absconv { e 1 . . . e n , f 1 . . . , f m } : f k ∈ S n − 1 } .

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work Random polytopes on the sphere (Gluskin’s polytopes) Given m ∈ N , we consider the set A m := { absconv { e 1 . . . e n , f 1 . . . , f m } : f k ∈ S n − 1 } . Note that we have the following mapping S n − 1 × · · · × S n − 1 → A m , given by ( f 1 , . . . , f m ) �→ absconv { e 1 . . . e n , f 1 . . . , f m } .

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work Random polytopes on the sphere (Gluskin’s polytopes) Given m ∈ N , we consider the set A m := { absconv { e 1 . . . e n , f 1 . . . , f m } : f k ∈ S n − 1 } . Note that we have the following mapping S n − 1 × · · · × S n − 1 → A m , given by ( f 1 , . . . , f m ) �→ absconv { e 1 . . . e n , f 1 . . . , f m } . This induces a measure ν in A m : the push-forward of the product measure µ n × µ n × · · · × µ n , where µ n is the probability surface measure on S n − 1 .

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work Carl/Pajor - Gluskin: If Z ∈ A m , then � log( m / n ) 1 n � vol ( Z ) . n

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work Carl/Pajor - Gluskin: If Z ∈ A m , then � log( m / n ) 1 n � vol ( Z ) . n 1 n � 1 In particular, if m ∼ n then vol ( Z ) n .

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work β B m = { Z ∈ A m : ∃ T ∈ SL n ( R ) with � T : X Z → X K � ≤ } . √ nvol ( K ) 1 n

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work β B m = { Z ∈ A m : ∃ T ∈ SL n ( R ) with � T : X Z → X K � ≤ } . √ nvol ( K ) 1 n G., Merzbacher, Pinasco: refinement of Khrabrov’s result Let K ⊂ R n be a centrally symmetric body. Then, 2 → X K �√ nvol ( K ) ν ( B m ) ≤ C n 2 ( � id : ℓ n 1 n ) n 2 β mn .

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work β B m = { Z ∈ A m : ∃ T ∈ SL n ( R ) with � T : X Z → X K � ≤ } . √ nvol ( K ) 1 n G., Merzbacher, Pinasco: refinement of Khrabrov’s result Let K ⊂ R n be a centrally symmetric body. Then, 2 → X K �√ nvol ( K ) ν ( B m ) ≤ C n 2 ( � id : ℓ n 1 n ) n 2 β mn . Suppose that for a given convex body K we have that 2 → X K �√ nvol ( K ) 1 ρ ( K ) := � id : ℓ n n is bounded by an absolute constant. Thus, ν ( B m ) ≤ D n 2 β mn ... By picking m ∼ n and β small enough we have ν ( B m ) < 1 .

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work β B m = { Z ∈ A m : ∃ T ∈ SL n ( R ) with � T : X Z → X K � ≤ } . √ nvol ( K ) 1 n G., Merzbacher, Pinasco: refinement of Khrabrov’s result Let K ⊂ R n be a centrally symmetric body. Then, 2 → X K �√ nvol ( K ) ν ( B m ) ≤ C n 2 ( � id : ℓ n 1 n ) n 2 β mn . Suppose that for a given convex body K we have that 2 → X K �√ nvol ( K ) 1 ρ ( K ) := � id : ℓ n n is bounded by an absolute constant. Thus, ν ( B m ) ≤ D n 2 β mn ... By picking m ∼ n and β small enough we have ν ( B m ) < 1 . Therefore, the complement is non-empty.

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work 1 n ∼ 1 Thus, there is some Z ∈ A cn (in particular vol ( Z ) n ) that is not in B cn . Note that in that case for every T ∈ SL n ( R ) we have β � T : X Z → X K � ≥ . √ nvol ( K ) 1 n

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work 1 n ∼ 1 Thus, there is some Z ∈ A cn (in particular vol ( Z ) n ) that is not in B cn . Note that in that case for every T ∈ SL n ( R ) we have β � T : X Z → X K � ≥ . √ nvol ( K ) 1 n Then, 1 vr( K , Z ) = vol ( K ) n · inf {� T : X Z → X K � : det( T ) = 1 } 1 vol ( Z ) n ≈ √ n β 1 n · ≥ nvol ( K ) √ nvol ( K ) 1 n

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work G., Merzbacher, Pinasco: refinement of Khrabrov’s result Let K ⊂ R n be a centrally symmetric body. Then, 2 → X K �√ nvol ( K ) 1 ν ( B m ) ≤ C n 2 ( � id : ℓ n n ) n 2 β mn . 2 → X K �√ nvol ( K ) 1 How to proceed in general if ρ ( K ) := � id : ℓ n n is not bounded by an absolute constant? Approximate the body K with a one that fulfills this hypothesis, without losing volume (the theory of isotropic convex bodies is involved).

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work How to approximate the body with a one with bounded ρ Given a centrally symmetric body K ⊂ R n there is W such that vr( W , K ) ≤ C 1 and 2 → X W �√ nvol ( W ) 1 ρ ( W ) = � id : ℓ n n ≤ C 2

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work How to approximate the body with a one with bounded ρ Given a centrally symmetric body K ⊂ R n there is W such that vr( W , K ) ≤ C 1 and 2 → X W �√ nvol ( W ) 1 ρ ( W ) = � id : ℓ n n ≤ C 2 Suppose that K ◦ is in isotropic position (so it has volume one).

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work How to approximate the body with a one with bounded ρ Given a centrally symmetric body K ⊂ R n there is W such that vr( W , K ) ≤ C 1 and 2 → X W �√ nvol ( W ) 1 ρ ( W ) = � id : ℓ n n ≤ C 2 Suppose that K ◦ is in isotropic position (so it has volume one). By a deep result of Paouris (2006) most of the mass of K ◦ concentrates inside √ nB n 2 .

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work How to approximate the body with a one with bounded ρ Given a centrally symmetric body K ⊂ R n there is W such that vr( W , K ) ≤ C 1 and 2 → X W �√ nvol ( W ) 1 ρ ( W ) = � id : ℓ n n ≤ C 2 Suppose that K ◦ is in isotropic position (so it has volume one). By a deep result of Paouris (2006) most of the mass of K ◦ concentrates inside √ nB n 2 . Take W ◦ := K ◦ ∩ √ nB n 1 2 . Observe that vol ( W ◦ ) n ∼ 1 so, 1 n ∼ 1 vol ( W ) n . On the other hand, vr( W , K ) ∼ vr( K ◦ , W ◦ ) ≤ C .

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work How to approximate the body with a one with bounded ρ Given a centrally symmetric body K ⊂ R n there is W such that vr( W , K ) ≤ C 1 and 2 → X W �√ nvol ( W ) 1 ρ ( W ) = � id : ℓ n n ≤ C 2 Suppose that K ◦ is in isotropic position (so it has volume one). By a deep result of Paouris (2006) most of the mass of K ◦ concentrates inside √ nB n 2 . Take W ◦ := K ◦ ∩ √ nB n 1 2 . Observe that vol ( W ◦ ) n ∼ 1 so, 1 n ∼ 1 vol ( W ) n . On the other hand, vr( W , K ) ∼ vr( K ◦ , W ◦ ) ≤ C . By construction, W ◦ ⊂ √ nB n 2 ⊂ √ nW (which 2 then B n 2 → X W � ≤ √ n ). implies that � id : ℓ n

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work Upper estimates...

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work Largest volume ratio for the cube Lemma (Dvoretzky-Rogers) Let L ⊂ R n be a convex body in John’s position, there are y 1 . . . y n ∈ Bd ( B n 2 ) ∩ Bd ( L ) such that � 1 � n − i + 1 2 � P span { y 1 ... y i − 1 } ⊥ ( y i ) � ≥ . n

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work Largest volume ratio for the cube Lemma (Dvoretzky-Rogers) Let L ⊂ R n be a convex body in John’s position, there are y 1 . . . y n ∈ Bd ( B n 2 ) ∩ Bd ( L ) such that � 1 � n − i + 1 2 � P span { y 1 ... y i − 1 } ⊥ ( y i ) � ≥ . n � n n � 1 det | y 1 . . . y n | = � y 1 � � n i =2 � P span { y 1 ... y i − 1 } ⊥ ( y i ) � ≥ 2 n !

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work Largest volume ratio for the cube

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work Largest volume ratio for the cube Consider the points y 1 , . . . , y n given by D-R. y 1 0 y 2

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work Largest volume ratio for the cube Consider the points y 1 , . . . , y n given by D-R. L is symmetric, so − y 2 y 1 − y 1 , . . . , − y n ∈ Bd ( L ) 0 − y 1 y 2

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work Largest volume ratio for the cube Consider the points y 1 , . . . , y n given by D-R. L is symmetric, so − y 2 y 1 − y 1 , . . . , − y n ∈ Bd ( L ) 0 P = ∩ n i =1 {� x , y i � ≤ 1 } − y 1 y 2 P

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work Largest volume ratio for the cube Consider the points y 1 , . . . , y n given by D-R. L is symmetric, so − y 2 y 1 − y 1 , . . . , − y n ∈ Bd ( L ) 0 P = ∩ n i =1 {� x , y i � ≤ 1 } − y 1 y 2 2 n det | y 1 ... y n | ≤ C n vol( P ) = P

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work Largest volume ratio for the cube Consider the points y 1 , . . . , y n given by D-R. L is symmetric, so − y 2 y 1 − y 1 , . . . , − y n ∈ Bd ( L ) 0 P = ∩ n i =1 {� x , y i � ≤ 1 } − y 1 y 2 2 n det | y 1 ... y n | ≤ C n vol( P ) = n ≤ √ n . 1 1 vol ( P ) n ≤ vol ( P ) n n 1 1 vol ( B n vol ( L ) 2 ) P

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work Largest volume ratio for the cube Consider the points y 1 , . . . , y n given by D-R. L is symmetric, so − y 2 y 1 − y 1 , . . . , − y n ∈ Bd ( L ) 0 P = ∩ n i =1 {� x , y i � ≤ 1 } − y 1 y 2 2 n det | y 1 ... y n | ≤ C n vol( P ) = n ≤ √ n . 1 1 vol ( P ) n ≤ vol ( P ) n n 1 1 vol ( B n vol ( L ) 2 ) P

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work Largest volume ratio for the cube Consider the points y 1 , . . . , y n given by D-R. L is symmetric, so − y 2 y 1 − y 1 , . . . , − y n ∈ Bd ( L ) 0 P = ∩ n i =1 {� x , y i � ≤ 1 } − y 1 y 2 2 n det | y 1 ... y n | ≤ C n vol( P ) = n ≤ √ n . 1 1 vol ( P ) n ≤ vol ( P ) n n 1 1 vol ( B n vol ( L ) 2 ) P ∞ ) ∼ √ n . lvr( B n

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work Unconditional bodies Definition A convex body K ⊂ R n is unconditional if ( x 1 , x 2 , . . . , x n ) ∈ L then ( ε 1 x 1 , ε 2 x 2 , . . . , ε n x n ) ∈ L for all ε i ∈ {− 1 , 1 } .

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work Unconditional bodies Definition A convex body K ⊂ R n is unconditional if ( x 1 , x 2 , . . . , x n ) ∈ L then ( ε 1 x 1 , ε 2 x 2 , . . . , ε n x n ) ∈ L for all ε i ∈ {− 1 , 1 } . Example unconditional: B n p for all 1 ≤ p ≤ ∞ , 2 ) ⊂ R n 2 . not unconditional: B L ( ℓ n

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work Unconditional bodies Definition A convex body K ⊂ R n is unconditional if ( x 1 , x 2 , . . . , x n ) ∈ L then ( ε 1 x 1 , ε 2 x 2 , . . . , ε n x n ) ∈ L for all ε i ∈ {− 1 , 1 } . Bobkov-Nazarov (2003) There is a constant c > 0 (independent of the dimension) such that for every unconditional isotropic convex body K ⊂ R n then [ − c , c ] n ⊂ K Sergey G Bobkov and Fedor L Nazarov. On convex bodies and log-concave probability measures with unconditional basis. In Geometric aspects of functional analysis , 2003.

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work Unconditional bodies Definition A convex body K ⊂ R n is unconditional if ( x 1 , x 2 , . . . , x n ) ∈ L then ( ε 1 x 1 , ε 2 x 2 , . . . , ε n x n ) ∈ L for all ε i ∈ {− 1 , 1 } . Bobkov-Nazarov (2003) There is a constant c > 0 (independent of the dimension) such that for every unconditional isotropic convex body K ⊂ R n then [ − c , c ] n ⊂ K Sergey G Bobkov and Fedor L Nazarov. On convex bodies and log-concave probability measures with unconditional basis. In Geometric aspects of functional analysis , 2003. vr( K , B n ∞ ) ≤ C .

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work If K ⊂ R n is unconditional lvr( K ) ∼ √ n .

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work If K ⊂ R n is unconditional lvr( K ) ∼ √ n . Indeed, ∞ , L ) ≤ C √ n . vr( K , L ) ≤ vr( K , B n ∞ )vr( B n

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work Largest volume ratio of the simplex The unit cube, C = [0 , 1] n .

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work Largest volume ratio of the simplex The unit cube, C = [0 , 1] n . C ⊂ S = co { 0 , ne 1 , . . . , ne n }

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work Largest volume ratio of the simplex The unit cube, C = [0 , 1] n . C ⊂ S = co { 0 , ne 1 , . . . , ne n } vol( C ) = 1 vol( S ) = n n n !

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work Largest volume ratio of the simplex The unit cube, C = [0 , 1] n . C ⊂ S = co { 0 , ne 1 , . . . , ne n } vol( C ) = 1 vol( S ) = n n n ! Applying Stirling’s formula: vr( S , B n ∞ ) = vr( S , C ) ≤ � vol ( S ) � 1 n ≈ 1 vol ( C )

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work Largest volume ratio of the simplex The unit cube, C = [0 , 1] n . C ⊂ S = co { 0 , ne 1 , . . . , ne n } vol( C ) = 1 vol( S ) = n n n ! Applying Stirling’s formula: vr( S , B n ∞ ) = vr( S , C ) ≤ � vol ( S ) � 1 n ≈ 1 vol ( C ) Therefore lvr( S ) ∼ √ n .

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work Schatten Classes Given A ∈ M d ( R ), consider s ( A ) = ( s 1 ( A ) , . . . , s d ( A )) the 1 sequence of eigenvalues of ( AA ∗ ) 2 . We define the p -Schatten norm on R d 2 as p = ( tr | A | p ) 1 / p , σ p ( A ) = � S ( A ) � ℓ d that is the ℓ p -norm of the singular values of A .

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work Schatten Classes Given A ∈ M d ( R ), consider s ( A ) = ( s 1 ( A ) , . . . , s d ( A )) the 1 sequence of eigenvalues of ( AA ∗ ) 2 . We define the p -Schatten norm on R d 2 as p = ( tr | A | p ) 1 / p , σ p ( A ) = � S ( A ) � ℓ d that is the ℓ p -norm of the singular values of A . p unit ball of the p -Schatten class in R d 2 . We denote B S d

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work G., Merzbacher, Pinasco For every 1 ≤ p ≤ ∞ , the largest volume ratio of the unit ball of the p -Schatten class (which is a set in R d 2 ) behaves as lvr( B S d p ) ∼ d .

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work G., Merzbacher, Pinasco For every 1 ≤ p ≤ ∞ , the largest volume ratio of the unit ball of the p -Schatten class (which is a set in R d 2 ) behaves as lvr( B S d p ) ∼ d . How?

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work G., Merzbacher, Pinasco For every 1 ≤ p ≤ ∞ , the largest volume ratio of the unit ball of the p -Schatten class (which is a set in R d 2 ) behaves as lvr( B S d p ) ∼ d . How? We give a very careful look at the proofs of the general upper inequalities.

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work G., Merzbacher, Pinasco For every 1 ≤ p ≤ ∞ , the largest volume ratio of the unit ball of the p -Schatten class (which is a set in R d 2 ) behaves as lvr( B S d p ) ∼ d . How? We give a very careful look at the proofs of the general upper inequalities. Again, all relies on the probabilistic method!!!!!!

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work Given T : X L → X K we have T ( L ) ⊂ � T � K and so 1 � T � vol ( K ) n vr( K , L ) ≤ . 1 1 n vol ( L ) (det T ) n

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work Given T : X L → X K we have T ( L ) ⊂ � T � K and so 1 � T � vol ( K ) n vr( K , L ) ≤ . 1 1 n vol ( L ) (det T ) n Chevet’s inequality If we denote O ( n ) the orthogonal group endowed with the Haar probability measure, then E � T : X L → X K � ≤ C √ n ( ℓ ( K ) � id : L → ℓ n 2 � 2 → X K � ℓ ( L ◦ )) . + � id : ℓ n

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work Given T : X L → X K we have T ( L ) ⊂ � T � K and so 1 � T � vol ( K ) n vr( K , L ) ≤ . 1 1 n vol ( L ) (det T ) n Chevet’s inequality If we denote O ( n ) the orthogonal group endowed with the Haar probability measure, then E � T : X L → X K � ≤ C √ n ( ℓ ( K ) � id : L → ℓ n 2 � 2 → X K � ℓ ( L ◦ )) . + � id : ℓ n We denote the right hand side of the inequality as α ( K , L ).

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work Given T : X L → X K we have T ( L ) ⊂ � T � K and so 1 � T � vol ( K ) n vr( K , L ) ≤ . 1 1 n vol ( L ) (det T ) n Chevet’s inequality If we denote O ( n ) the orthogonal group endowed with the Haar probability measure, then E � T : X L → X K � ≤ C √ n ( ℓ ( K ) � id : L → ℓ n 2 � 2 → X K � ℓ ( L ◦ )) . + � id : ℓ n We denote the right hand side of the inequality as α ( K , L ). 1 vr( K , L ) ≤ α ( K , L ) vol ( K ) n . 1 vol ( L ) n

The volume ratio Lower estimates Upper bounds: some particular cases Future/Recent work Idea: Since the volume ratio is affinely invariant we look for a position of L such that � id : L → ℓ n 2 � and ℓ ( L ◦ ) are “not so big”. 1 n is “not so small”. vol ( L ) Proposition: For every centrally symmetric convex body K ⊂ R n , 1 lvr( K ) ≤ C � id : ℓ n n n log( n ) . 2 → X K � vol ( K )