curvature bounds: discrete versus continuous spacesanca bonciocat, imar bucharest curvature bounds:...

ROUGH CURVATURE BOUNDS ROUGH CD-CONDITION FIN

Curvature bounds: discrete versus continuousspaces

Anca Bonciocat, IMAR Bucharest

Based on a joint work with K. T. Sturm

ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES

Approach

1 based on mass transportation, following

K. T. Sturm, On the geometry of metric measure spaces I,II, Acta Math. 2006

J. Lott, C. Villani, Ricci curvature for metric-measurespaces via optimal transport, Ann. of Math. 2009

2 rough curvature bounds will depend on a real parameterh > 0

Approach

Point of view - Coarse geometry

Coarse geometry studies the "large scale" properties of spaces.

(M, d,m) metric measure space(M, d) is a complete separable metric spacem is a measure on (M,B(M)), which is locally finite in thesense that m(Br (x)) <∞ for all x ∈ M and all sufficientlysmall r > 0.

We say that the metric measure space (M, d,m) is normalizedif m(M) = 1.

(M, d,m) and (M ′, d′,m′) are isomorphic iff there exists anisometry ψ : M0 → M ′

0 between the supportsM0 := supp[m] ⊂ M and M ′

0 := supp[m′] ⊂ M ′ such thatψ∗m = m′.

The diameter of a metric measure space (M, d,m) will be thediameter of the metric space (supp[m],d).

The L2-Wasserstein space

The L2-Wasserstein distance between two measures µ and νon M is defined as

dW (µ, ν) = inf

{(∫M×M

d2(x , y)dq(x , y)

: q coupling of µ, ν

with the convention inf ∅ = ∞.

P2(M, d) :={ν :∫

M d2(o, x)dν(x) <∞ for some o ∈ M}

(P2(M, d), dW ) is called L2-Wasserstein space over (M, d).

P2(M, d,m) := {ν ∈ P2(M, d) : ν � m}.

dW (µ, ν) = inf

{(∫M×M

d2(x , y)dq(x , y)

P2(M, d) :={ν :∫

P2(M, d,m) := {ν ∈ P2(M, d) : ν � m}.

dW (µ, ν) = inf

{(∫M×M

d2(x , y)dq(x , y)

P2(M, d) :={ν :∫

P2(M, d,m) := {ν ∈ P2(M, d) : ν � m}.

dW (µ, ν) = inf

{(∫M×M

d2(x , y)dq(x , y)

P2(M, d) :={ν :∫

P2(M, d,m) := {ν ∈ P2(M, d) : ν � m}.

dW (µ, ν) = inf

{(∫M×M

d2(x , y)dq(x , y)

P2(M, d) :={ν :∫

P2(M, d,m) := {ν ∈ P2(M, d) : ν � m}.

The relative entropy

Ent(ν|m) :=

M ρ log ρdm , for ν = ρ ·m

+∞ , otherwise

We denote by P∗2(M, d,m) the subspace of measuresν ∈ P2(M, d,m) of finite entropy Ent(ν|m) <∞.

The relative entropy

Ent(ν|m) :=

M ρ log ρdm , for ν = ρ ·m

+∞ , otherwise

We denote by P∗2(M, d,m) the subspace of measuresν ∈ P2(M, d,m) of finite entropy Ent(ν|m) <∞.

Riemannian case

Theorem (v.Renesse-Sturm 2005)

For any smooth connected Riemannian manifold M withintrinsic metric d and volume measure m and any K ∈ R thefollowing properties are equivalent :

1 Ricx(v , v) ≥ K |v |2 for x ∈ M and v ∈ Tx(M).2 The entropy Ent(·|m) is displacement K -convex on P2(M)

in the sense that for each geodesic γ : [0,1] → P2(M) andfor each t ∈ [0,1]

Ent(γ(t)|m) ≤ (1− t)Ent(γ(0)|m) + tEnt(γ(1)|m)

t(1− t) d2W (γ(0), γ(1)).

Riemannian case

Theorem (v.Renesse-Sturm 2005)

For any smooth connected Riemannian manifold M withintrinsic metric d and volume measure m and any K ∈ R thefollowing properties are equivalent :

1 Ricx(v , v) ≥ K |v |2 for x ∈ M and v ∈ Tx(M).2 The entropy Ent(·|m) is displacement K -convex on P2(M)

in the sense that for each geodesic γ : [0,1] → P2(M) andfor each t ∈ [0,1]

Ent(γ(t)|m) ≤ (1− t)Ent(γ(0)|m) + tEnt(γ(1)|m)

t(1− t) d2W (γ(0), γ(1)).

Curvature bounds for metric measure spaces

Definition (Sturm, Acta Math. 2006)

A metric measure space (M, d,m) has curvature ≥ K for somenumber K ∈ R iff the relative entropy Ent(·|m) is weaklyK -convex on P∗2(M, d,m) in the sense that for each pairν0, ν1 ∈ P∗2(M, d,m) there exists a geodesicΓ : [0,1] → P∗2(M, d,m) connecting ν0 and ν1 with

Ent(Γ(t)|m) ≤ (1− t)Ent(Γ(0)|m) + tEnt(Γ(1)|m)

t(1− t) d2W (Γ(0), Γ(1))

for all t ∈ [0,1].

Curvature bounds for metric measure spaces

A metric measure space (M, d,m) has curvature ≥ K for somenumber K ∈ R iff the relative entropy Ent(·|m) is weaklyK -convex on P∗2(M, d,m) in the sense that for each pairν0, ν1 ∈ P∗2(M, d,m) there exists a geodesicΓ : [0,1] → P∗2(M, d,m) connecting ν0 and ν1 with

Ent(Γ(t)|m) ≤ (1− t)Ent(Γ(0)|m) + tEnt(Γ(1)|m)

t(1− t) d2W (Γ(0), Γ(1))

for all t ∈ [0,1].

L2-transportation distance D

D((M, d,m), (M ′, d′,m′)) = inf(∫

MtM′d̂

2(x , y)dq(x , y)

where d̂ ranges over all couplings of d and d′ and q rangesover all couplings of m and m′.

A pseudo-metric d̂ on the disjoint union M tM ′ is a coupling ofd and d′ if d̂(x , y) = d(x , y) and d̂(x ′, y ′) = d′(x ′, y ′) for allx , y ∈ supp[m] ⊂ M and all x ′, y ′ ∈ supp[m′] ⊂ M ′.

D defines a complete separable length metric on the family ofall isomorphism classes of normalized metric measure spaces(M, d,m) with m ∈ P2(M, d).

D((M, d,m), (M ′, d′,m′)) = inf(∫

MtM′d̂

2(x , y)dq(x , y)

D((M, d,m), (M ′, d′,m′)) = inf(∫

MtM′d̂

2(x , y)dq(x , y)

D((M, d,m), (M ′, d′,m′)) = inf(∫

MtM′d̂

2(x , y)dq(x , y)

Let h > 0 be given. We say that a metric space (M, d) ish-rough geodesic iff for each pair of points x0, x1 ∈ M and eacht ∈ [0,1] there exists a point xt ∈ M satisfying

d(x0, xt) ≤ t d(x0, x1) + h

d(xt , x1) ≤ (1− t) d(x0, x1) + h

The point xt will be referred to as the h-rough t-intermediatepoint between x0 and x1.

d(x0, xt) ≤ t d(x0, x1) + h

d(xt , x1) ≤ (1− t) d(x0, x1) + h

d(x0, xt) ≤ t d(x0, x1) + h

d(xt , x1) ≤ (1− t) d(x0, x1) + h

Examples

1 Any nonempty set X with the discrete metric d(x , y) = 0for x = y and 1 for x 6= y is h-rough geodesic for anyh ≥ 1/2. In this case, any point is an h-midpoint of any pairof distinct points.

2 If ε > 0 then the space (Rn, d) with the metricd(x , y) = |x − y | ∧ ε is h-rough geodesic for h ≥ ε/2 (here|·| is the euclidian metric).

3 For ε > 0 the space (Rn, d) with the metricd(x , y) =

√ε|x − y |+ |x − y |2 is h-rough geodesic for

each h ≥ ε/4.4 Discrete spaces, graphs.

Examples

Let (M, d) be a metric space. For each h > 0 and any pair ofmeasures ν0, ν1 ∈ P2(M, d) put

d±hW (ν0, ν1) := inf

{(∫[( d(x0, x1)∓h)+]2 dq(x0, x1)

)1/2},

where q ranges over all couplings of ν0 and ν1 and (·)+ denotesthe positive part.

The infimum above is attained. A coupling q for which theinfimum is attained in the definition of d±h

W is called ±h-optimalcoupling.

Let (M, d) be a metric space. For each h > 0 and any pair ofmeasures ν0, ν1 ∈ P2(M, d) put

d±hW (ν0, ν1) := inf

{(∫[( d(x0, x1)∓h)+]2 dq(x0, x1)

)1/2},

where q ranges over all couplings of ν0 and ν1 and (·)+ denotesthe positive part.

The infimum above is attained. A coupling q for which theinfimum is attained in the definition of d±h

W is called ±h-optimalcoupling.

LemmaFor any 0 < h and 0 < h1 < h2 we have :

1 d+hW ≤ dW ≤ d+h

W + h ;2 dW ≤ d−h

W ≤ dW + h

3 d−h1W < d−h2

4 d+h1W (ν0, ν1) ≥ d+h2

W (ν0, ν1) with strict inequality if and onlyif d+h1

W (ν0, ν1) > 0.

Rough curvature bounds

Definition

(M, d,m) has h-rough curvature ≥ K iff for each pairν0, ν1 ∈ P∗2(M, d,m) and for any t ∈ [0,1] there exists anh-rough t-intermediate point ηt ∈ P∗2(M, d,m) between ν0 andν1 satisfying

Ent(ηt |m) ≤ (1−t)Ent(ν0|m)+tEnt(ν1|m)−K2

t(1−t) d±hW (ν0, ν1)

where the sign in d±hW (ν0, ν1) is chosen ’+’ if K > 0 and ’−’ if

K < 0.

Briefly, we write in this case h- Curv(M, d,m) ≥ K .

Definition

t(1−t) d±hW (ν0, ν1)

K < 0.

Definition

t(1−t) d±hW (ν0, ν1)

K < 0.

Properties

1 If (M, d,m) and (M ′, d′,m′) are isomorphic then

h- Curv(M, d,m) ≥ K ⇔ h- Curv(M ′, d′,m′) ≥ K

2 For 0 < h1 < h2

h1- Curv(M, d,m) ≥ K ⇒ h2- Curv(M, d,m) ≥ K .

3 If (M, d,m) is a metric measure space and α, β > 0 then

h- Curv(M, d,m) ≥ K ⇔ αh- Curv(M, α d, βm) ≥ Kα2

Properties

2 For 0 < h1 < h2

Properties

2 For 0 < h1 < h2

Properties

2 For 0 < h1 < h2

Properties

2 For 0 < h1 < h2

Stability under convergence

TheoremLet (M, d,m) be a compact normalized metric measure spaceand consider {(Mh, dh,mh)}h>0 a family of normalized metricmeasure spaces with uniformly bounded diameter and withh- Curv(Mh, dh,mh) ≥ Kh for Kh → K as h → 0. If

(Mh, dh,mh)D−→ (M, d,m)

as h → 0 thenCurv(M, d,m) ≥ K .

TheoremLet (M, d,m) be a compact normalized metric measure spaceand consider {(Mh, dh,mh)}h>0 a family of normalized metricmeasure spaces with uniformly bounded diameter and withh- Curv(Mh, dh,mh) ≥ Kh for Kh → K as h → 0. If

as h → 0 thenCurv(M, d,m) ≥ K .

Discretizations of metric measure spaces

Let (M, d,m) be a given metric measure space.

For h > 0 let Mh = {xn : n ∈ N}, with M =∞⋃

i=1BR(xi), where

R = R(h) ↘ 0 as h ↘ 0.

Choose Ai ⊂ BR(xi) mutually disjoint with xi ∈ Ai , i = 1,2, . . .

and∞⋃

i=1Ai = M.

Define mh({xi}) := m(Ai), i = 1,2, . . . measure on Mh.

We call (Mh, d,mh) a discretization of (M, d,m).

i=1BR(xi), where

R = R(h) ↘ 0 as h ↘ 0.

and∞⋃

i=1Ai = M.

i=1BR(xi), where

R = R(h) ↘ 0 as h ↘ 0.

and∞⋃

i=1Ai = M.

i=1BR(xi), where

R = R(h) ↘ 0 as h ↘ 0.

and∞⋃

i=1Ai = M.

i=1BR(xi), where

R = R(h) ↘ 0 as h ↘ 0.

and∞⋃

i=1Ai = M.

i=1BR(xi), where

R = R(h) ↘ 0 as h ↘ 0.

and∞⋃

i=1Ai = M.

Theorem

1 If m(M) <∞ then (Mh, d,mh)D−→ (M, d,m) as h → 0.

2 If Curv(M, d,m) ≥ K for some real number K then foreach h > 0 and for each discretization (Mh, d,mh) withR(h) ≤ h/4 we have h- Curv(Mh, d,mh) ≥ K .

Examples

1 Zn with the metric d1 coming from the norm|x |1 =

∑ni=1 |xi | and with the measure mn =

∑x∈Zn δx has

h- Curv(Zn, d1,mn) ≥ 0 for any h ≥ 2n.2 The n-dimensional grid En equipped with the graph

distance and with the measure mn which is the1-dimensional Lebesgue measure on the edges, hash- Curv(En, d1,mn) ≥ 0 for any h ≥ 2(n + 1).

3 G the graph that tiles the euclidian plane with equilateraltriangles of edge r , with the graph metric, and with the1-dimensional Lebesgue measure on the edges, hash-curvature ≥ 0 for any h ≥ 8r

√3/3.

4 G′ the graph that tiles the euclidian plane with regularhexagons of edge length r , with the graph metric and withthe 1-dimensional measure m′, has h-curvature ≥ 0 for anyh ≥ 34r/3.

Examples

∑x∈Zn δx has

√3/3.

Examples

∑x∈Zn δx has

√3/3.

Examples

∑x∈Zn δx has

√3/3.

Examples

∑x∈Zn δx has

√3/3.

Homogeneous planar graphs

G(l ,n, r) with vertices of constant degree l ≥ 3, with facesbounded by polygons with n ≥ 3 edges and with all edges ofthe same length r > 0.

1 If 1l + 1

n <12 then G(l ,n, r) can be embedded into the

2-dimensional hyperbolic space with constant sectionalcurvature

K = − 1r2

[arccosh

cos2 (πn

)sin2 (π

) − 1

2 If 1l + 1

n >12 then G(l ,n, r) can be embedded into the

2-dimensional sphere with constant sectional curvature

K =1r2

[arccos

cos2 (πn

)sin2 (π

) − 1

3 If 1l + 1

n = 12 then G(l ,n, r) can be embedded into the

euclidian plane (K = 0).

1 If 1l + 1

K = − 1r2

[arccosh

cos2 (πn

)sin2 (π

) − 1

2 If 1l + 1

K =1r2

[arccos

cos2 (πn

)sin2 (π

) − 1

3 If 1l + 1

1 If 1l + 1

K = − 1r2

[arccosh

cos2 (πn

)sin2 (π

) − 1

2 If 1l + 1

K =1r2

[arccos

cos2 (πn

)sin2 (π

) − 1

3 If 1l + 1

We equip G(l ,n, r) with the metric d induced by thecorresponding Riemannian metric and with the uniformmeasure m on the edges.

Proposition

(G(l ,n, r), d,m) has h-curvature ≥ K for h ≥ r · C(l ,n), where

− 1r2

[arccosh

cos2(πn )

sin2(πl )− 1)]2

for 1l + 1

[arccos

cos2(πn )

sin2(πl )− 1)]2

for 1l + 1

0 for 1l + 1

n = 12

We equip G(l ,n, r) with the metric d induced by thecorresponding Riemannian metric and with the uniformmeasure m on the edges.

Proposition

(G(l ,n, r), d,m) has h-curvature ≥ K for h ≥ r · C(l ,n), where

− 1r2

[arccosh

cos2(πn )

sin2(πl )− 1)]2

for 1l + 1

[arccos

cos2(πn )

sin2(πl )− 1)]2

for 1l + 1

0 for 1l + 1

n = 12

Here C(l ,n) = 4 ·arcsinh

sin(πn )

scos2(π

sin2(πl )−1

arccosh

2cos2(π

sin2(πl )−1

Transportation cost inequality

The probability measure m satisfies a Talagrand inequality (ortransportation cost inequality) with constant K iff for allν ∈ P2(M, d)

dW (ν,m) ≤√

2 Ent(ν|m)

Proposition ("h-Talagrand Inequality")

Assume that (M, d,m) is a metric measure space which hash- Curv(M, d,m) ≥ K for some numbers h > 0 and K > 0.Then for each ν ∈ P2(M, d) we have

d+hW (ν,m) ≤

√2 Ent(ν|m)

K. (1.1)

We will call (1.1) h-Talagrand inequality.ANCA BONCIOCAT, IMAR BUCHAREST CURVATURE BOUNDS: DISCRETE VERSUS CONTINUOUS SPACES

dW (ν,m) ≤√

2 Ent(ν|m)

d+hW (ν,m) ≤

√2 Ent(ν|m)

K. (1.1)

dW (ν,m) ≤√

2 Ent(ν|m)

d+hW (ν,m) ≤

√2 Ent(ν|m)

K. (1.1)

dW (ν,m) ≤√

2 Ent(ν|m)

d+hW (ν,m) ≤

√2 Ent(ν|m)

K. (1.1)

Concentration of measure

For a given A ⊂ M measurable denoteBr (A) := {x ∈ M : d(x ,A) < r} for r > 0.

The concentration function of (M, d,m) is defined as

α(M,d,m)(r) := sup{

1−m(Br (A)) : A ∈ B(M),m(A) ≥ 12

}, r > 0.

Proposition

Let (M, d,m) be a metric measure space with h- Curv(M, d,m)≥ K > 0 for some h > 0. Then there exists an r0 > 0 such thatfor all r ≥ r0

α(M, d,m)(r) ≤ e−Kr2/8.

1−m(Br (A)) : A ∈ B(M),m(A) ≥ 12

}, r > 0.

Proposition

α(M, d,m)(r) ≤ e−Kr2/8.

1−m(Br (A)) : A ∈ B(M),m(A) ≥ 12

}, r > 0.

Proposition

α(M, d,m)(r) ≤ e−Kr2/8.

1−m(Br (A)) : A ∈ B(M),m(A) ≥ 12

}, r > 0.

Proposition

α(M, d,m)(r) ≤ e−Kr2/8.

1−m(Br (A)) : A ∈ B(M),m(A) ≥ 12

}, r > 0.

Proposition

α(M, d,m)(r) ≤ e−Kr2/8.

The Rényi entropy functional

SN(·|m) : P2(M, d) → R

withSN(ν|m) := −

∫Mρ−1/Ndν,

where ρ is the density of the absolutely continuous part νc withrespect to m in the Lebesgue decompositionν = νc + νs = ρm + νs of the measure ν ∈ P2(M, d).

SN(·|m) : P2(M, d) → R

withSN(ν|m) := −

∫Mρ−1/Ndν,

where ρ is the density of the absolutely continuous part νc withrespect to m in the Lebesgue decompositionν = νc + νs = ρm + νs of the measure ν ∈ P2(M, d).

Assume that m(M) is finite.

1 For each N > 1 the Rényi entropy functional SN(·|m) islower semicontinuous and satisfies

−m(M)1/N ≤ SN(·|m) ≤ 0 on P2(M, d).

2 For any ν ∈ P2(M, d)

Ent(·|m) = limN→∞

N(1 + SN(ν|m)).

−m(M)1/N ≤ SN(·|m) ≤ 0 on P2(M, d).

N(1 + SN(ν|m)).

−m(M)1/N ≤ SN(·|m) ≤ 0 on P2(M, d).

N(1 + SN(ν|m)).

−m(M)1/N ≤ SN(·|m) ≤ 0 on P2(M, d).

N(1 + SN(ν|m)).

For given K ,N ∈ R with N ≥ 1 and (t , θ) ∈ [0,1]× R+ put

τ(t)K ,N(θ) =

∞ , if K θ2 ≥ (N − 1)π2

(sin“q

KN−1 tθ

”sin“q

KN−1 θ

”)1− 1

, if 0 < K θ2 < (N − 1)π2

t , if K θ2 = 0 orif K θ2 < 0 and N = 1

“q−KN−1 tθ

”sinh

“q−KN−1 θ

”)1− 1

, if K θ2 < 0 and N > 1

Curvature-dimension condition for metric spaces

(M, d,m) satisfies the curvature-dimension condition CD(K ,N)iff for each pair ν0, ν1 ∈ P2(M, d,m) there exist an optimalcoupling q of ν0, ν1 and a geodesic Γ : [0,1] → P2(M, d,m)connecting ν0, ν1 and with

SN′(ηt |m) ≤ −∫ [

τ(1−t)K ,N′ ( d(x0, x1)) · ρ

−1/N′

0 (x0)

+ τ(t)K ,N′( d(x0, x1)) · ρ

−1/N′

1 (x1)]

dq(x0, x1)

for all t ∈ [0,1] and all N ′ ≥ N. Here ρi denotes the densityfunctions of the absolutely continuous parts of νi with respect tom, i = 1,2.

Curvature-dimension condition for metric spaces

(M, d,m) satisfies the curvature-dimension condition CD(K ,N)iff for each pair ν0, ν1 ∈ P2(M, d,m) there exist an optimalcoupling q of ν0, ν1 and a geodesic Γ : [0,1] → P2(M, d,m)connecting ν0, ν1 and with

SN′(ηt |m) ≤ −∫ [

τ(1−t)K ,N′ ( d(x0, x1)) · ρ

−1/N′

0 (x0)

+ τ(t)K ,N′( d(x0, x1)) · ρ

−1/N′

1 (x1)]

dq(x0, x1)

for all t ∈ [0,1] and all N ′ ≥ N. Here ρi denotes the densityfunctions of the absolutely continuous parts of νi with respect tom, i = 1,2.

The rough curvature-dimension condition

Let (M, d) be a metric space and h ≥ 0, t ∈ [0,1] given realnumbers.

1 xt is an h-rough t-intermediate point of x0 and x1 in M if{d(x0, xt) ≤ t d(x0, x1) + hd(xt , x1) ≤ (1− t) d(x0, x1) + h

2 xt is an h-rough t-intermediate point of x0 and x1 in thestrong sense if

(1− t) d(x0, xt)2 + t d(xt , x1)

2 ≤ t(1− t) d(x0, x1)2 + h2.

(1− t) d(x0, xt)2 + t d(xt , x1)

2 ≤ t(1− t) d(x0, x1)2 + h2.

(1− t) d(x0, xt)2 + t d(xt , x1)

2 ≤ t(1− t) d(x0, x1)2 + h2.

(1− t) d(x0, xt)2 + t d(xt , x1)

2 ≤ t(1− t) d(x0, x1)2 + h2.

Definition

(M, d,m) satisfies h-CD(K ,N) (resp. h-CDs(K ,N)) iff for eachpair ν0, ν1 ∈ P2(M, d,m) there exists a δh-optimal coupling q ofν0, ν1 such that for any t ∈ [0,1] there exists an h-rought-intermediate point (resp. in the strong sense)ηt ∈ P2(M, d,m) of ν0, ν1 with

SN′(ηt |m) ≤ −∫ [

τ(1−t)K ,N′ (( d(x0, x1)−δh)+) · ρ−1/N′

0 (x0)

+τ(t)K ,N′(( d(x0, x1)−δh)+) · ρ−1/N′

1 (x1)]

dq(x0, x1)

for all N ′ ≥ N.

ρi = the density of the absolutely continuous part of νi w.r.t. m.δ = −1 for K < 0 and δ = 1 for K ≥ 0

Definition

SN′(ηt |m) ≤ −∫ [

τ(1−t)K ,N′ (( d(x0, x1)−δh)+) · ρ−1/N′

0 (x0)

+τ(t)K ,N′(( d(x0, x1)−δh)+) · ρ−1/N′

1 (x1)]

dq(x0, x1)

Definition

SN′(ηt |m) ≤ −∫ [

τ(1−t)K ,N′ (( d(x0, x1)−δh)+) · ρ−1/N′

0 (x0)

+τ(t)K ,N′(( d(x0, x1)−δh)+) · ρ−1/N′

1 (x1)]

dq(x0, x1)

Definition

SN′(ηt |m) ≤ −∫ [

τ(1−t)K ,N′ (( d(x0, x1)−δh)+) · ρ−1/N′

0 (x0)

+τ(t)K ,N′(( d(x0, x1)−δh)+) · ρ−1/N′

1 (x1)]

dq(x0, x1)

For K = 0 the above inequality reads

SN′(ηt |m) ≤ (1− t) · SN′(ν0|m) + t · SN′(ν1|m),

h-CD(0,N) requires the Rényi entropy functionals SN′(·|m) tobe weakly convex on P2(M, d,m) along "h-rough geodesics" forall N ′ ≥ N.

For K = 0 the above inequality reads

SN′(ηt |m) ≤ (1− t) · SN′(ν0|m) + t · SN′(ν1|m),

h-CD(0,N) requires the Rényi entropy functionals SN′(·|m) tobe weakly convex on P2(M, d,m) along "h-rough geodesics" forall N ′ ≥ N.

The rough curvature-dimension condition - Properties

Suppose that (M, d,m) satisfies the h-CD(K ,N) condition.Then :

1 (M, d,m) satisfies h′-CD(K ′,N ′) for any K ′ ≤ K , N ′ ≥ Nand h′ ≥ h.

2 Any metric space (M ′, d′,m′) isomorphic to (M, d,m)satisfies the same h-CD(K ,N).

3 For any α, β > 0 the metric measure space (M, α d, βm)satisfies the αh-CD(α−2K ,N) condition.

4 If (M, d,m) has finite mass then h- Curv(M, d,m) ≥ K .Therefore, h- Curv(M, d,m) ≥ K may be seen as a roughcurvature-dimension condition h-CD(K ,∞).

The same holds true of we replace everywhere h-CD(K ,N) byh-CDs(K ,N).

Geometrical consequences

(M, d,m) metric measure space

µ0, µ1 probability measures, Ai := supp[µi ]

η h-rough t-intermediate point in the strong sense of µ0 and µ1

For λ ≥ 0 denote

Aλt :=

{y ∈ M : ∃(x0, x1) ∈ A0 × A1 with (1− t) d(x0, y)2

+t d(y , x1)2 ≤ t(1− t)( d(x0, x1)

2 + λ2)}.

Then the following estimate holds :

η({Aλt ) ≤ h2/λ2 for any λ > 0.

For λ ≥ 0 denote

Aλt :=

{y ∈ M : ∃(x0, x1) ∈ A0 × A1 with (1− t) d(x0, y)2

+t d(y , x1)2 ≤ t(1− t)( d(x0, x1)

2 + λ2)}.

For λ ≥ 0 denote

Aλt :=

{y ∈ M : ∃(x0, x1) ∈ A0 × A1 with (1− t) d(x0, y)2

+t d(y , x1)2 ≤ t(1− t)( d(x0, x1)

2 + λ2)}.

For λ ≥ 0 denote

Aλt :=

{y ∈ M : ∃(x0, x1) ∈ A0 × A1 with (1− t) d(x0, y)2

+t d(y , x1)2 ≤ t(1− t)( d(x0, x1)

2 + λ2)}.

For λ ≥ 0 denote

Aλt :=

{y ∈ M : ∃(x0, x1) ∈ A0 × A1 with (1− t) d(x0, y)2

+t d(y , x1)2 ≤ t(1− t)( d(x0, x1)

2 + λ2)}.

For λ ≥ 0 denote

Aλt :=

{y ∈ M : ∃(x0, x1) ∈ A0 × A1 with (1− t) d(x0, y)2

+t d(y , x1)2 ≤ t(1− t)( d(x0, x1)

2 + λ2)}.

Moreover, if 0 = λ0 ≤ λ1 ≤ λ2 ≤ . . . ≤ λi ≤ . . . then

∞∑i=1

λ2i · η(A

λi+1t \ Aλi

t ) ≤ h2

or, equivalently,

∞∑i=1

η({Aλit )(λ2

i − λ2i−1) ≤ h2.

Moreover, if 0 = λ0 ≤ λ1 ≤ λ2 ≤ . . . ≤ λi ≤ . . . then

∞∑i=1

λ2i · η(A

λi+1t \ Aλi

t ) ≤ h2

or, equivalently,

∞∑i=1

η({Aλit )(λ2

i − λ2i−1) ≤ h2.

Geometrical consequences - Brunn-Minkowski ineq.

The classical Brunn-Minkowski in Rn :

voln(tA + (1− t)B)1/n ≥ t voln(A)1/n + (1− t)voln(B)1/n

for all bounded Borel measurable subsets A and B and anyt ∈ [0,1].

The classical Brunn-Minkowski in Rn :

voln(tA + (1− t)B)1/n ≥ t voln(A)1/n + (1− t)voln(B)1/n

for all bounded Borel measurable subsets A and B and anyt ∈ [0,1].

Proposition

Let (M, d,m) be a metric measure space that has finite massand satisfies h-CDs(K ,N) for some numbers h ≥ 0, K ,N ∈ R,N ≥ 1. Then for any measurable sets A0, A1 ⊂ M withm(A0) ·m(A1) > 0, for any t ∈ [0,1] , N ′ ≥ N and any λ > 0

m(Aλt )1/N′

+(h2/λ2)1−1/N′m({Aλ

t )1/N′ ≥ τ(1−t)K ,N′ (Θh) ·m(A0)

1/N′

+τ(t)K ,N′(Θh) · (A1)

1/N′,

with Aλt as before and Θh is given by

Θh :=

{infx0∈A0,a1∈A1( d(x0, x1)− h)+, if K ≥ 0supx0∈A0,a1∈A1

( d(x0, x1) + h), if K < 0.

Proposition

m(Aλt )1/N′

+(h2/λ2)1−1/N′m({Aλ

t )1/N′ ≥ τ(1−t)K ,N′ (Θh) ·m(A0)

1/N′

+τ(t)K ,N′(Θh) · (A1)

1/N′,

Θh :=

( d(x0, x1) + h), if K < 0.

Proposition

m(Aλt )1/N′

+(h2/λ2)1−1/N′m({Aλ

t )1/N′ ≥ τ(1−t)K ,N′ (Θh) ·m(A0)

1/N′

+τ(t)K ,N′(Θh) · (A1)

1/N′,

Θh :=

( d(x0, x1) + h), if K < 0.

Corollary (’Generalized Brunn-Minkowski Inequality’)

Assume that (M, d,m) is a normalized metric measure spacethat satisfies h-CDs(K ,N). Then for any measurable sets A0,A1 ⊂ M with m(A0) ·m(A1) > 0, for any t ∈ [0,1] and N ′ ≥ N

m(A√

ht )1/N′

+h1−1/N′ ≥ τ(1−t)K ,N′ (Θh)m(A0)

1/N′+τ

(t)K ,N′(Θh)m(A1)

1/N′,

with Θh given above.In particular, if K ≥ 0 then

m(A√

ht )1/N′

+h1−1/N′ ≥ (1− t) ·m(A0)1/N′

+ t ·m(A1)1/N′

Corollary (’Generalized Brunn-Minkowski Inequality’)

Assume that (M, d,m) is a normalized metric measure spacethat satisfies h-CDs(K ,N). Then for any measurable sets A0,A1 ⊂ M with m(A0) ·m(A1) > 0, for any t ∈ [0,1] and N ′ ≥ N

m(A√

ht )1/N′

+h1−1/N′ ≥ τ(1−t)K ,N′ (Θh)m(A0)

1/N′+τ

(t)K ,N′(Θh)m(A1)

1/N′,

with Θh given above.In particular, if K ≥ 0 then

m(A√

ht )1/N′

+h1−1/N′ ≥ (1− t) ·m(A0)1/N′

+ t ·m(A1)1/N′

Geometrical consequences - Bonnet-Myers Theorem

Corollary

For every normalized metric measure space (M, d,m) thatsatisfies the rough curvature-dimension condition h-CDs(K ,N)for some real numbers h > 0, K > 0 and N ≥ 1, the support ofthe measure m has diameter

L ≤√

N − 1K

In particular, if K > 0 and N = 1 then supp[m] consists of a ballof radius h.

TheoremLet (M, d,m) be a normalized metric measure space and{(Mh, dh,mh)}h>0 a family of normalized metric measurespaces such that for each h > 0 the space (Mh, dh,mh)satisfies h-CD(Kh,Nh) and has diameter Lh for some realnumbers Kh,Nh and Lh with Nh ≥ 1 and Lh > 0. Assume that

and(Kh,Nh,Lh) → (K ,N,L)

as h → 0 for some (K ,N,L) ∈ R3 satisfying K · L2 < (N − 1)π2.Then the space (M, d,m) satisfies the curvature-dimensioncondition CD(K ,N) and has diameter ≤ L.

TheoremLet (M, d,m) be a normalized metric measure space and{(Mh, dh,mh)}h>0 a family of normalized metric measurespaces such that for each h > 0 the space (Mh, dh,mh)satisfies h-CD(Kh,Nh) and has diameter Lh for some realnumbers Kh,Nh and Lh with Nh ≥ 1 and Lh > 0. Assume that

and(Kh,Nh,Lh) → (K ,N,L)

as h → 0 for some (K ,N,L) ∈ R3 satisfying K · L2 < (N − 1)π2.Then the space (M, d,m) satisfies the curvature-dimensioncondition CD(K ,N) and has diameter ≤ L.

Stability under discretization

TheoremLet (M, d,m) be a metric measure space that satisfies thecurvature-dimension condition CD(K ,N) for some realnumbers K and N ≥ 1. Then for each h > 0 any discretization(Mh, d,mh) with R(h) ≤ h/4 satisfies the roughcurvature-dimension condition h-CD(K ,N).

Thank you for your attention

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