$\forall (\xi,\eta),$

Introduction: the MTW tensor

$$\mathfrak{S}_c(x,y)(\xi,\eta)=(c_{ik\bar m} c^{\bar m n} c_{n \bar\jmath\bar\ell}-c_{i\bar\jmath k\bar\ell})\xi^i\eta^{\bar\jmath}\xi^k\eta^{\bar\ell}$$

$X,Y,$ $d$-dimensional manifolds,   $c\in C^4(X\times Y)$,   $\nabla^2_{xy}c(x,y)$ nonsingular

MTW conditions

MTW tensor

(Ma–Trudinger–Wang ’05)

$$\mathfrak{S}_c(x,y)(\xi,\eta)\geq \kappa \lvert\xi\rvert_x^2 \lvert\eta\rvert_{x,y}^2$$

$\kappa>0$

$\kappa=0$

(Ma–Trudinger–Wang ’05)

(Trudinger–Wang ’09)

$$\mathfrak{S}_c(x,y)(\xi,\eta)\geq0$$

(Kim–McCann ’10)

$$\mathfrak{S}_c(x,y)(\xi,\eta)\geq \kappa \lvert\xi\rvert_x^2 \lvert\eta\rvert_{x,y}^2-C\lvert\langle \xi,\eta\rangle\rvert \lvert\xi\rvert\lvert\eta\rvert$$

(Loeper–Villani ’10)

⏵  $\forall (\xi,\eta),\,\,\xi\perp\eta$

Nonnegative cross-curvature (NNCC)

⏵  $\forall (\xi,\eta),$

⏵  

Motivation 1: principal–agents problems

$\Psi^c=$ set of $c$-concave functions

$$\phi^c(x)\coloneqq\min_{y\in Y} c(x,y)+\phi(y)$$

$\Psi^c$ is a convex set (of functions)

T H E O R E M   (FIGALLI–KIM–MCCANN '11)

If $c$ has NNCC + other assumptions then

$$T_\phi(x)\coloneqq\operatorname*{argmin}_{y\in Y}\,\_\_\_\_\_\_\_\_\_$$

$(X,d)$ metric space

Convergence rate

EVI

$(X,d)$ non-positively curved, Mayer/Jost

Ambrosio–Gigli–Savaré

$\mathcal{E}$ convex on geodesics

$\mathcal{E}$ convex on curves $x(t)$ such that $d^2(x,x_{n-1})$ is $1$-convex, i.e. $t\mapsto d^2(x(t),x_{n-1})-t^2 \,d^2(x(1),x(0))$ is convex

$t\mapsto (x(t),\bar y)$ is a c-segment if $$\nabla_yc(x(t),\bar y)=(1-t)\nabla_yc(x(0),\bar y)+t\nabla_yc(x(1),\bar y).$$

D E F I N I T I O N

Background

$X,Y$ two $d$-dimensional manifolds, 

$c\in C^4(X\times Y)$, nondegenerate, twist,

$X\times Y$ c-convex

T H E O R E M   (KIM–MCCANN '10)

$c$ has nonnegative cross-curvature iff for any c-segment $t\mapsto (x(t),\bar y)$, $$\forall y\in Y,\quad t\mapsto c(x(t),\bar y)-c(x(t),y)\text{    is convex}.$$ 

Key observation

$c\in C^1(X\times Y$. If $x(t)$ is any curve such that $t\mapsto c(x(t),\bar y)-c(x(t),y)$  is convex for every $y\in Y$, then $(x(t),\bar y)$ is a c-segment.

D E F I N I T I O N  (L–Todeschi–Vialard '24)

$(X\times Y,c)$ is an NNCC space if for each $(x_0,x_1,\bar y)\in X\times X\times Y$, there exists a path $x(\cdot)$ from $x_0$ to $x_1$ such that $\forall y\in Y$, $$c(x(t),\bar y)-c(x(t),y)\leq (1-t)[c(x_0,\bar y)-c(x_0,y)]+t[c(x_1,\bar y)-c(x_1,y)].$$

$(x(t),\bar y) \eqqcolon$ generalized c-segment.

$X, Y$ two arbitrary sets,   $c\colon X\times Y\to\mathbb{R}\cup\{+\infty\}$.

(L–Todeschi–Vialard '24)

Some properties

Products preserve NNCC

If $(X^1\times Y^1,c^1)$ and $(X^2\times Y^2,c^2)$ are NNCC then so is $((X^1\times X^2)\times(Y^1\times Y^2),c)$ with $$c((x^1,x^2),(y^1,y^2))=c^1(x^1,y^1)+c^2(x^2,y^2).$$

Certain projections preserve NNCC

$(X\times Y,c)$ NNCC space,   $\pi_1\colon X\to\underline{X}$,   $\pi_2\colon Y\to\underline{Y}$.$$\underline{c}(\underline{x},\underline{y})=\inf\{c(x,y) : \pi_1(x)=\underline{x},\pi_2(y)=\underline{y}\}$$ Additional assumptions. Then $(\underline{X}\times \underline{Y},\underline{c})$ is NNCC

$(\mathcal{P}_2(\mathbb{R}^n)\times \mathcal{P}_2(\mathbb{R}^n), W_2^2)$ is an NNCC space.

More generally $X$, $Y$ Polish spaces, $c\in C(X\times Y)$.

If $(X\times Y,c)$ is an NNCC space then so is $(\mathcal{P}(X)\times \mathcal{P}(Y), \mathcal{T}_c)$.

T H E O R E M   (L–Todeschi–Vialard '24)

The Wasserstein space is NNCC

A generalized c-segment for $\mathcal{T}_c$ between $(\mu_0,\nu)$ and $(\mu_1,\nu)$ is given by $(\mu(t),\nu)$

  ⏵ $(T_0,S)$ optimal coupling of $(\mu_0,\nu)$ 

  ⏵ $(T_1,S)$ optimal coupling of $(\mu_1,\nu)$

  ⏵ for each $\omega\in\Omega$, $t\mapsto (T_t(\omega),S(\omega))$ is a $c$-segment

  ⏵ $\mu(t)=(T_t)_\#\mathbb{P}$

$(\Omega,\mathcal{F},\mathbb{P})$

$T_0,T_1\colon\Omega\to X$

$S\colon\Omega\to Y$

More examples of NNCC spaces

Bures–Wasserstein

Gromov–Wasserstein        $\mathbf{X}=[X,f,\mu]$   and   $\mathbf{Y}=[Y,g,\nu]$

$$\operatorname{GW}^2(\mathbf{X},\mathbf{Y})=\inf_{\pi\in\Pi(\mu,\nu)}\int\lvert f(x,x')-g(y,y')\rvert^2\,d\pi(x,y)\,d\pi(x',y')\,.$$

Unbalanced OT

Application: EVIs

$c\colon X\times X\to\mathbb{R}_+$,   $\mathcal{E}\colon X\to\mathbb{R}$

Convergence rates:

EVI

$c=W_2^2, \,\,\mathcal{T}_c,\,\, \operatorname{GW}^2,\,\, \operatorname{BW}^2\dots$

⏵ $(X\times X,c)$ NNCC space 

⏵ $\mathcal{E}$ convex on generalized c-segments $(x(t),x_{n-1})$

⏵ $c$ satisfies $$\liminf_{t\to 0}\frac{c(x(t),x(0))}{t}=0.$$

Then EVI.

T H E O R E M   (L–Todeschi–Vialard '24)

EVI

Thank you!