Geometric inequalities on the Heisenberg group Kinga Sipos - PowerPoint PPT Presentation

Geometric inequalities on the Heisenberg group Geometric inequalities on the Heisenberg group Kinga Sipos University of Bern kinga.sipos@math.unibe.ch MAnET Midterm Meeting, Helsinki, 8-9 December 2015

Geometric inequalities on the Heisenberg group Definition of the Brunn-Minkowski, Pr´ ekopa-Leindler and Borell-Brascamp-Lieb inequalities 1 Definition of the Brunn-Minkowski, Pr´ ekopa-Leindler and Borell-Brascamp-Lieb inequalities 2 Proof of the Borell-Brascamp-Lieb inequality in the euclidean case Proof for normalized functions f , g and p = − 1 n 3 Borell-Brascamp-Lieb inequality on the Riemannian manifolds 4 N. Juillet’s disproval of existence for some types of Brunn-Minkowski inequality 5 Further ideas / possibilities 6 Bibliography MAnET Midterm Meeting, Helsinki, 8-9 December 2015 2

Geometric inequalities on the Heisenberg group Definition of the Brunn-Minkowski, Pr´ ekopa-Leindler and Borell-Brascamp-Lieb inequalities ...in the euclidean case Brunn-Minkowski inequality | (1 − s ) A + sB | 1 / n ≥ (1 − s ) | A | 1 / n + s | B | 1 / n for all A , B ⊂ R n Borel sets, s ∈ [0 , 1] Pr´ ekopa-Leindler inequality Let f , g , h : R n → [0 , ∞ ) be measurable functions and fix s ∈ (0 , 1). h ((1 − s ) x + sy ) ≥ f ( x ) 1 − s g ( y ) s , ∀ x , y ∈ R n ⇒ � 1 − s �� � s � �� R n h ≥ R n f R n g Borell-Brascamp-Lieb inequality Let f , g , h : R n → [0 , ∞ ) be measurable functions, fix s ∈ (0 , 1) and p ≥ − 1 n . p s ( f ( x ) , g ( y )) , ∀ x , y ∈ R n ⇒ h ((1 − s ) x + sy ) ≥ M p � 1+ np �� � � h ≥ M f , g , s s ( a , b ) = ((1 − s ) a p + sb p ) 1 / p , for any a , b > 0, p ∈ R \ { 0 } and where M p s ( a , b ) = a 1 − s b s , which is obtained from M p s ∈ [0 , 1] and M 0 s ( a , b ) by p → 0. MAnET Midterm Meeting, Helsinki, 8-9 December 2015 3

Geometric inequalities on the Heisenberg group Definition of the Brunn-Minkowski, Pr´ ekopa-Leindler and Borell-Brascamp-Lieb inequalities Relation between the BM, PL and BBL inequalities Observe that: Borell-Brascamp-Lieb inequality ⇒ Pr´ ekopa-Leindler inequality 1 p → 0 ((1 − s ) a p + sb p ) p → 0 M p As M 0 p = a 1 − s b s , for all s ( a , b ) = lim s ( s , b ) = lim a , b > 0 and s ∈ [0 , 1], PL can be obtained by setting p = 0 in BBL. Borell-Brascamp-Lieb inequality ⇒ Brunn-Minkowski inequality Choosing f , g and h to be the characteristic functions of the Borel sets A , B , respectively Z s ( A , B ), these functions satisfy the condition of the BBL inequality, which implies that | Z s ( A , B ) | 1 / n ≥ (1 − s ) | A | 1 / n + s | B | 1 / n . MAnET Midterm Meeting, Helsinki, 8-9 December 2015 4

Geometric inequalities on the Heisenberg group Definition of the Brunn-Minkowski, Pr´ ekopa-Leindler and Borell-Brascamp-Lieb inequalities ...in case of the Heisenberg group How to define the intermediate points? (Let s ∈ [0 , 1] be fixed.) in the euclidean case for the s -intermediate point associated to the pointpair ( x , y ) ∈ R n × R n we use the convex combination (1 − s ) x + sy MAnET Midterm Meeting, Helsinki, 8-9 December 2015 5

Geometric inequalities on the Heisenberg group Definition of the Brunn-Minkowski, Pr´ ekopa-Leindler and Borell-Brascamp-Lieb inequalities ...in case of the Heisenberg group How to define the intermediate points? (Let s ∈ [0 , 1] be fixed.) in the euclidean case for the s -intermediate point associated to the pointpair ( x , y ) ∈ R n × R n we use the convex combination (1 − s ) x + sy with the Heisenberg group operator ( ∗ ) and λ -dilation ( ρ λ ) an s -intermediate point associated to the pointpair ( x , y ) ∈ H n × H n can be defined as ρ 1 − s ( x ) ∗ ρ s ( y ) MAnET Midterm Meeting, Helsinki, 8-9 December 2015 6

Geometric inequalities on the Heisenberg group Definition of the Brunn-Minkowski, Pr´ ekopa-Leindler and Borell-Brascamp-Lieb inequalities ...in case of the Heisenberg group How to define the intermediate points? (Let s ∈ [0 , 1] be fixed.) in the euclidean case for the s -intermediate point associated to the pointpair ( x , y ) ∈ R n × R n we use the convex combination (1 − s ) x + sy with the Heisenberg group operator ( ∗ ) and λ -dilation ( ρ λ ) an s -intermediate point associated to the pointpair ( x , y ) ∈ H n × H n can be defined as ρ 1 − s ( x ) ∗ ρ s ( y ) with the help of geodesics an s -intermediate point between x ∈ H n and y ∈ H n can be defined as that point on the geodesic connecting the two points, which divides the geodesic in segments with ratio s : (1 − s ) MAnET Midterm Meeting, Helsinki, 8-9 December 2015 7

Geometric inequalities on the Heisenberg group Definition of the Brunn-Minkowski, Pr´ ekopa-Leindler and Borell-Brascamp-Lieb inequalities ... in case of the Heisenberg group When y ∈ cut ( x ), the geodesic from x to y is not uniquely defined. Let’s introduce the notation Z s ( x , y ) for the set of s -intermediate points associated to ( x , y ) ∈ H n × H n : Z s ( x , y ) = { z ∈ H n | d ( x , z ) = sd ( x , y ) and d ( z , y ) = (1 − s ) d ( x , y ) } For A , B ⊂ H n define � Z s ( A , B ) = Z s ( x , y ) ( x , y ) ∈ A × B MAnET Midterm Meeting, Helsinki, 8-9 December 2015 8

Geometric inequalities on the Heisenberg group Definition of the Brunn-Minkowski, Pr´ ekopa-Leindler and Borell-Brascamp-Lieb inequalities ...in case of the Heisenberg group Brunn-Minkowski inequality | Z s ( A , B ) | 1 / d ≥ (1 − s ) | A | 1 / d + s | B | 1 / d for all A , B ⊂ H n Borel sets, s ∈ [0 , 1] Pr´ ekopa-Leindler inequality Let f , g , h : R n → [0 , ∞ ) be measurable functions and fix s ∈ (0 , 1). � 1 − s �� � s �� h ( z ) ≥ f ( x ) 1 − s g ( y ) s , ∀ x , y ∈ H n , z ∈ Z s ( x , y ) ⇒ � H n h ≥ H n f H n g Borell-Brascamp-Lieb inequality Let f , g , h : H n → [0 , ∞ ) be measurable functions, fix s ∈ (0 , 1) and p ≥ − 1 d . h ( z ) ≥ M p s ( f ( x ) , g ( y )) , ∀ x , y ∈ H n , z ∈ Z s ( x , y ) ⇒ p �� � � 1+ dp � ⇒ H n h ≥ M H n f , H n g s MAnET Midterm Meeting, Helsinki, 8-9 December 2015 9

Geometric inequalities on the Heisenberg group Definition of the Brunn-Minkowski, Pr´ ekopa-Leindler and Borell-Brascamp-Lieb inequalities ... in case of the Heisenberg group How to choose d ? Use for d the topological demension 2 n + 1? the homogenous dimension 2 n + 2? something else? MAnET Midterm Meeting, Helsinki, 8-9 December 2015 10

Geometric inequalities on the Heisenberg group Proof of the Borell-Brascamp-Lieb inequality in the euclidean case Sketch Proof for normalized functions f , g and p = − 1 n . Borell-Brascamp-Lieb inequality for normalized functions Let f , g , h : R n → [0 , ∞ ) be measurable functions with � � R n f = R n g = 1. Fix s ∈ (0 , 1). � − 1 ( f ( x ) , g ( y )) , ∀ x , y ∈ R n ⇒ h ((1 − s ) x + sy ) ≥ M h ≥ 1 n s R n Rescaling argument. MAnET Midterm Meeting, Helsinki, 8-9 December 2015 11

Geometric inequalities on the Heisenberg group Proof of the Borell-Brascamp-Lieb inequality in the euclidean case Proof for normalized functions f , g and p = − 1 n Notation: supp ( f ) = X , supp ( g ) = Y , µ = f d x , ν = g d y . Consider a convex function ϕ : R n → R such that for S = ▽ ϕ : X → Y , S # µ = ν . Consider the displacement interpolant measure of µ and ν : [ µ, ν ] s = ( S s )# µ with probability density ρ s , where S s = (1 − s ) Id + s ▽ ϕ . By the concavity of the det ( · ) 1 / n function over symmetric, positive-semidefinite matrices det ((1 − s ) I n + sHess ( ϕ ( x ))) 1 / n ≥ (1 − s )( det ( I n )) 1 / n + s ( Hess ( ϕ ( x ))) 1 / n . Monge-Amp` ere for f and ρ s : f ( x ) = ρ s ( S s ( x )) Jac ( S s )( x ) , µ − a . e x , where Jac ( S s )( x ) = det ((1 − s ) I n + sHess ϕ ( x )). Monge-Amp` ere for f and g : f ( x ) = g ( S ( x )) Jac ( S )( x ) , µ − a . e x , where Jac ( S )( x ) = det ( Hess ( ϕ ( x ))). MAnET Midterm Meeting, Helsinki, 8-9 December 2015 12

Geometric inequalities on the Heisenberg group Proof of the Borell-Brascamp-Lieb inequality in the euclidean case Proof for normalized functions f , g and p = − 1 n det ((1 − s ) I n + sHess ( ϕ ( x ))) 1 / n ≥ (1 − s )( det ( I n )) 1 / n + s ( Hess ( ϕ ( x ))) 1 / n f ( x ) = ρ s ( S s ( x )) det ((1 − s ) I n + sHess ϕ ( x )) f ( x ) = g ( S ( x )) det ( Hess ( ϕ ( x ))) MAnET Midterm Meeting, Helsinki, 8-9 December 2015 13

Geometric inequalities on the Heisenberg group Proof of the Borell-Brascamp-Lieb inequality in the euclidean case Proof for normalized functions f , g and p = − 1 n det ((1 − s ) I n + sHess ( ϕ ( x ))) 1 / n ≥ (1 − s )( det ( I n )) 1 / n + s ( Hess ( ϕ ( x ))) 1 / n f ( x ) = ρ s ( S s ( x )) det ((1 − s ) I n + sHess ϕ ( x )) f ( x ) = g ( S ( x )) det ( Hess ( ϕ ( x ))) � f ( x ) � 1 / n � 1 / n � f ( x ) ⇒ ≥ (1 − s ) + s ρ s ( x ) g ( S ( x )) (1 − s )( f ( x )) − 1 / n + sg ( S ( x )) − 1 / n ( ρ s ( x )) − 1 / n ≥ M − 1 / n ρ s ( x ) ≤ ( f ( x ) , g ( S ( x ))) s MAnET Midterm Meeting, Helsinki, 8-9 December 2015 14