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 ⊂ Rn Borel sets, s ∈ [0, 1] Pr´ ekopa-Leindler inequality Let f , g, h : Rn → [0, ∞) be measurable functions and fix s ∈ (0, 1). h((1 − s)x + sy) ≥ f (x)1−sg(y)s, ∀x, y ∈ Rn ⇒

• Rn h ≥
• Rn f

1−s

Rn g

s Borell-Brascamp-Lieb inequality Let f , g, h : Rn → [0, ∞) be measurable functions, fix s ∈ (0, 1) and p ≥ − 1

n.

h((1 − s)x + sy) ≥ Mp

s (f (x), g(y)), ∀x, y ∈ Rn ⇒

• h ≥ M

p 1+np

s

• f ,
• g
• ,

where Mp

s (a, b) = ((1 − s)ap + sbp)1/p, for any a, b > 0, p ∈ R \ {0} and

s ∈ [0, 1] and M0

s (a, b) = a1−sbs, which is obtained from Mp 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 As M0

s (a, b) = lim p→0 Mp s (s, b) = lim p→0((1 − s)ap + sbp)

1 p = a1−sbs, for all

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 Zs(A, B), these functions satisfy the condition of the BBL inequality, which implies that |Zs(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) ∈ Rn × Rn 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) ∈ Rn × Rn we use the convex combination (1 − s)x + sy with the Heisenberg group operator (∗) and λ-dilation (ρλ) an s-intermediate point associated to the pointpair (x, y) ∈ Hn × Hn can be defined as ρ1−s(x) ∗ ρs(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) ∈ Rn × Rn we use the convex combination (1 − s)x + sy with the Heisenberg group operator (∗) and λ-dilation (ρλ) an s-intermediate point associated to the pointpair (x, y) ∈ Hn × Hn can be defined as ρ1−s(x) ∗ ρs(y) with the help of geodesics an s-intermediate point between x ∈ Hn and y ∈ Hn 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 Zs(x, y) for the set of s-intermediate points associated to (x, y) ∈ Hn × Hn: Zs(x, y) = {z ∈ Hn|d(x, z) = sd(x, y) and d(z, y) = (1 − s)d(x, y)} For A, B ⊂ Hn define Zs(A, B) =

• (x,y)∈A×B

Zs(x, y)

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 |Zs(A, B)|1/d ≥ (1 − s)|A|1/d + s|B|1/d for all A, B ⊂ Hn Borel sets, s ∈ [0, 1] Pr´ ekopa-Leindler inequality Let f , g, h : Rn → [0, ∞) be measurable functions and fix s ∈ (0, 1). h(z) ≥ f (x)1−sg(y)s, ∀x, y ∈ Hn, z ∈ Zs(x, y) ⇒

• Hn h ≥
• Hn f

1−s

Hn g

s Borell-Brascamp-Lieb inequality Let f , g, h : Hn → [0, ∞) be measurable functions, fix s ∈ (0, 1) and p ≥ − 1

d . h(z) ≥ Mp s (f (x), g(y)), ∀x, y ∈ Hn, z ∈ Zs(x, y) ⇒

⇒

• Hn h ≥ M

p 1+dp

s

• Hn f ,
• Hn g
• 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 2n + 1? the homogenous dimension 2n + 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 : Rn → [0, ∞) be measurable functions with

• Rn f =
• Rn g = 1.

Fix s ∈ (0, 1). h((1 − s)x + sy) ≥ M

− 1

n

s

(f (x), g(y)), ∀x, y ∈ Rn ⇒

• Rn

h ≥ 1 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 dx, ν = gdy. Consider a convex function ϕ : Rn → R such that for S = ▽ϕ : X → Y , S#µ = ν. Consider the displacement interpolant measure of µ and ν: [µ, ν]s = (Ss)#µ with probability density ρs, where Ss = (1 − s)Id + s▽ϕ. By the concavity of the det(·)1/n function over symmetric, positive-semidefinite matrices det((1−s)In+sHess(ϕ(x)))1/n ≥ (1−s)(det(In))1/n+s(Hess(ϕ(x)))1/n. Monge-Amp` ere for f and ρs: f (x) = ρs(Ss(x))Jac(Ss)(x), µ − a.e x, where Jac(Ss)(x) = det((1 − s)In + sHessϕ(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)In +sHess(ϕ(x)))1/n ≥ (1−s)(det(In))1/n +s(Hess(ϕ(x)))1/n f (x) = ρs(Ss(x))det((1 − s)In + sHessϕ(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)In +sHess(ϕ(x)))1/n ≥ (1−s)(det(In))1/n +s(Hess(ϕ(x)))1/n f (x) = ρs(Ss(x))det((1 − s)In + sHessϕ(x)) f (x) = g(S(x))det(Hess(ϕ(x))) ⇒ f (x) ρs(x) 1/n ≥ (1 − s) + s

• f (x)

g(S(x)) 1/n (ρs(x))−1/n ≥ (1 − s)(f (x))−1/n + sg(S(x))−1/n ρs(x) ≤ M−1/n

s

(f (x), g(S(x)))

MAnET Midterm Meeting, Helsinki, 8-9 December 2015 14

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)In +sHess(ϕ(x)))1/n ≥ (1−s)(det(In))1/n +s(Hess(ϕ(x)))1/n f (x) = ρs(Ss(x))det((1 − s)In + sHessϕ(x)) f (x) = g(S(x))det(Hess(ϕ(x))) ⇒ f (x) ρs(x) 1/n ≥ (1 − s) + s

• f (x)

g(S(x)) 1/n (ρs(x))−1/n ≥ (1 − s)(f (x))−1/n + sg(S(x))−1/n ρs(x) ≤ M−1/n

s

(f (x), g(S(x))) ≤ h(Zs(x, S(x))) µ − a.e x ⇓

• Rn h ≥ 1

MAnET Midterm Meeting, Helsinki, 8-9 December 2015 15

Geometric inequalities on the Heisenberg group Borell-Brascamp-Lieb inequality on the Riemannian manifolds

On the Riemannian manifold M, the optimal transport map pushing µ forward to ν can be written, as S(x) = expx(−▽ϕ(x)), where ϕ is a c-concave function relative to the support of f and g. Ss can be defined in point x as expx(−s▽ϕ(x)). The Jacobian determinant of Ss satisfies the following inequality for µ − a.e. x (Jac(Ss)(x))

1 n ≥ (1 − s) (v1−s(S(x), x)) 1 n + s (vt(x, S(x))) 1 n (Jac(S)(x)) 1 n ,

where the appearing volume distortion coefficient is defined as vs(x, y) = lim

r→0

vol(Zt(x, B(y, r)) vol(B(y, sr)) , ∀x, y ∈ M and s ∈ (0, 1].

MAnET Midterm Meeting, Helsinki, 8-9 December 2015 16

Geometric inequalities on the Heisenberg group Borell-Brascamp-Lieb inequality on the Riemannian manifolds

Borell-Brascamp-Lieb inequality on the Riemannian manifolds Let M be a continuously curved, n-dimensional Riemannian manifold. Fix p ≥ − 1

n and s ∈ (0, 1). Let f , g, h : M → [0, ∞) be three measurable

functions, A and B two Borel sets of M such that suppf ⊂ A, suppg ⊂ B and assume that h(z) ≥ Mp

s

• f (x)

v1−s(y, x), g(y) vs(x, y)

• for all (x, y) ∈ A × B, z ∈ Zs(x, y).

Then

• M

h ≥ M

p 1+np

s

• M

f ,

• M

g

• .

MAnET Midterm Meeting, Helsinki, 8-9 December 2015 17

Geometric inequalities on the Heisenberg group

• N. Juillet’s disproval of existence for some types of Brunn-Minkowski inequality
• N. Juillet disproves the existence of

geodesic Brunn-Minkowski inequality in any dimension. mutiplicative Brunn-Minkowski for any dimension less than the topogical dimension of the Heisenberg group.

MAnET Midterm Meeting, Helsinki, 8-9 December 2015 18

Geometric inequalities on the Heisenberg group Further ideas / possibilities

Bad news: In the Heisenberg group the optimal transport map doesn’t assure the existence of such a determinant inequality that we used for the proof in the euclidean case and in the riemannian case, therefore to prove the Broell-Brascamp-Lieb inequality the same approach cannot be used. Juillet’s disproval Good news: We know from D. Cordero-Erasquin, R. J. McCann, M. Schmuckenschl¨ ager, that on the Riemannian manifolds a weighted Borell-Brascamp-Lieb inequality holds.

• L. Ambrosio and S. Rigot consider a Riemannian approximation for the

Heisenberg group from the point of view of the optimal mass transportation approach. Using these two results we try to formulate a Borell-Brascap-Lieb inequality on the Heisenberg group, which can be obtained as a limit Riemannian Borell-Brascap-Lieb inequalities.

MAnET Midterm Meeting, Helsinki, 8-9 December 2015 19

Geometric inequalities on the Heisenberg group Bibliography

References I

• L. Ambrosio, S. Rigot

Optimal mass transportation in the Heisenberg group, Journal of Functional Analysis Volume 208, Issue 2 (2004), 261-301.

• D. Cordero-Erasquin, R. J. McCann, M. Schmuckenschl¨

ager A Riemannian interpolation inequality ` a la Borell, Brascamp and Lieb, Inventiones mathematicae 146 (2), (2001), 219-257.

• N. Juillet

Geometric Inequalities and Generalized Ricci Bounds in the Heisenberg Group, Int Math Res Notices (2009), vol. 13

• C. Villani

Topics in Optimal Transportation, AMS (2003).

MAnET Midterm Meeting, Helsinki, 8-9 December 2015 20

Geometric inequalities on the Heisenberg group Bibliography

Thank you!

MAnET Midterm Meeting, Helsinki, 8-9 December 2015 21