A remark on the geometric interpretation of the A3w condition from optimal transport

In optimal transport a condition known as A3w is necessary for regularity of the optimal transport map. Here we provide a geometric interpretation of A3w. We’ll use freely the notation from [4]. Let $c\in C^{2}(\mathbf{R}^{n}\times\mathbf{R}^{n})$ satisfy A1 and A2 (see §2). Keeping in mind the prototypical case $c(x,y)=|x-y|^{2}$, we fix $x_{0},y_{0}\in\mathbf{R}^{n}$ and perform a linear transformation so $c_{xy}(x_{0},y_{0})=-I$. Define coordinates

and the inverse transformations by $x(q),y(p)$. Write $c(q,p)=c(x(q),y(p))$ and let $q_{0}=q(x_{0})$ and $p_{0}=p(y_{0})$. We prove A3w is satisfied if and only if whenever these transformations are performed

Heuristically, A3w implies when $q-q_{0}$ “points in the same direction” as $p-p_{0}$ it is cheaper to transport $q$ to $p$ and $q_{0}$ to $p_{0}$ than the alternative $q$ to $p_{0}$ and $q_{0}$ to $p$. Thus, A3w implies compatibility between directions in the cost-convex geometry and the cost of transport.

A3w first appeared (in a stronger form) in [4]. It was weakened in [6] and a new interpretation given in [2]. The impetus for the above interpretation is Lemma 2.1 in [1]. Our result can also be realised by a particular choice of c-convex function in the unpublished preprint [5].

Acknowledgements. My thanks to Jiakun Liu and Robert McCann for helpful comments and discussion.

Let $c\in C^{2}(\mathbf{R}^{n}\times\mathbf{R}^{n})$ satisfy following the well known conditions
A1. For each $x_{0},y_{0}\in\mathbf{R}^{n}$ the following mappings are injective

A2. For each $x_{0},y_{0}\in\mathbf{R}^{n}$ we have $\det c_{i,j}(x_{0},y_{0})\neq 0$.

Here, and throughout, subscripts before a comma denote differentiation with respect to the first variable, subscripts after a comma denote differentiation with respect to the second variable.

By A1 we define on $\mathcal{U}:=\{(x,c_{x}(x,y));x,y\in\mathbf{R}^{n}\}$ a mapping $Y:\mathcal{U}\rightarrow\mathbf{R}^{n}$ by

The A3w condition, usually expressed with fourth derivatives but written here as in [3], is the following.
A3w. Fix $x$. The function

is concave along line segments orthogonal to $\xi$.

To verify A3w it suffices to verify midpoint concavity, that is whenever $\xi\cdot\eta=0$ there holds

Finally, we recall $A\subset\mathbf{R}^{n}$ is called $c$-convex with respect to $y_{0}$ provided $c_{y}(A,y_{0})$ is convex. When A3w is satisfied and $y,y_{0}\in\mathbf{R}^{n}$ are given the section $\{x\in\mathbf{R}^{n};c(x,y)>c(x,y_{0})\}$ is $c$-convex with respect to $y_{0}$ [3].

Now fix $(x_{0},p_{0})\in\mathcal{U}$ and $y_{0}=Y(x_{0},p_{0})$. To simplify the proof we assume $x_{0},y_{0},q_{0},p_{0}=0$. Up to an affine transformation (replace $y$ with $\tilde{y}:=-c_{xy}(0,0)y$) we assume $c_{xy}(0,0)=-I$. Note with $q,p$ as defined in (1), (2), this implies $\frac{\partial q}{\partial x}(0)=I$. Put