# Strict $g$-convexity for generated Jacobian equations with applications to
global regularity

Cale Rankin<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract.

This article has two purposes. The first is to prove solutions of the second
boundary value problem for generated Jacobian equations (GJEs) are strictly
$g$-convex. The second is to prove the global $C^{3}$ regularity of
Aleksandrov solutions to the same problem. In particular, Aleksandrov
solutions are classical solutions. These are related because the strict
$g$-convexity is essential for the proof of the global regularity. The
assumptions for the strict $g$-convexity are the natural extension of those
used by Chen and Wang in the optimal transport case. They are the Loeper
maximum principle condition, a positively pinched right-hand side, a
$g^{*}$-convex target, and a source domain strictly contained in a $g$-convex
domain. This improves the existing domain conditions though at the expense of
requiring a $C^{3}$ generating function. This is appropriate for global
regularity where existence is proved assuming a $C^{4}$ generating function.
We prove the global regularity under the hypothesis that Jiang and Trudinger
recently used to obtain the existence of a globally smooth solution and an
additional condition on the height of solutions. Our proof of global
regularity is by modifying Jiang and Trudinger’s existence result to construct
a globally $C^{3}$ solution intersecting the Aleksandrov solution. Then the
strict convexity yields the interior regularity to apply the author’s
uniqueness results.

This research is supported by an Australian Government Research Training
Program (RTP) Scholarship and by ARC DP 200101084.

##  1\. Introduction

Generated Jacobian equations generalise the Monge–Ampère equation to encompass
applications to geometric optics. For these equations a generalised notion of
convexity, known as $g$-convexity, plays the same role as convexity does for
Monge–Ampère equations. In this article we prove the strict $g$-convexity of
solutions to the second boundary value problem for GJEs. As an application of
this result we prove the global $C^{3}$ regularity of Aleksandrov solutions to
the same problem under stronger hypothesis.

Generated Jacobian equations are PDE of the form

(1) $\det DY(\cdot,u,Du)=\psi(\cdot,u,Du)\text{ in }\Omega,$

where
$Y:\mathbf{R}^{n}\times\mathbf{R}\times\mathbf{R}^{n}\rightarrow\mathbf{R}^{n}$
is a vector field with a particular structure: It arises from a generating
function as outlined in Section 2. The other components of (1) are a
nonnegative function
$\psi:\mathbf{R}^{n}\times\mathbf{R}\times\mathbf{R}^{n}\rightarrow\mathbf{R}$
and a domain $\Omega\subset\mathbf{R}^{n}$. For the strict $g$-convexity we
assume

(2) $\lambda\leq\psi(\cdot,u,Du)\leq\Lambda,$

for positive constants $\lambda,\Lambda$. For the global regularity we assume

(3) $\psi(\cdot,u,Du)=\frac{f(\cdot)}{f^{*}(Y(\cdot,u,Du))},$

for $C^{2}$ positive functions $f,f^{*}$. In each case (1) is coupled with the
second boundary value problem:

(4) $Y(\cdot,u,Du)(\Omega)=\Omega^{*},$

for a prescribed domain $\Omega^{*}.$ We require convexity conditions on
$\Omega,\Omega^{*}$. These, and other required definitions and structure
conditions, are introduced in Section 2 where we also state our main results:
Theorems 1 and 2.

Generated Jacobian equations were introduced by Trudinger [29] to extend the
theory of Monge–Ampère type equations in optimal transport to problems in
geometric optics. Thus, to situate our results we outline the optimal
transport case.

The Monge–Ampère equation, recovered from (1) by taking $Y(x,u,Du)=Du$, is a
fully nonlinear PDE, elliptic when $u$ is convex. Brenier [1] showed that a
(suitably defined) weak solution of (1) subject to (4) is obtained by solving
the optimal transport problem with quadratic cost. Thus studying the
regularity of optimal transport maps is tantamount to studying the
Monge–Ampère equation paired with the second boundary value problem. The work
relevant to ours is that of Caffarelli [3, 2] and Urbas [33]. Caffarelli’s
work takes place under the assumption (2) and is concerned with the strict
convexity and $C^{1,\alpha}$ regularity of a weak notion of solution, known as
Aleksandrov solutions. Urbas, in the smooth setting, obtained the global
estimates required for the method of continuity, thereby proving the existence
of globally regular solutions. The uniqueness of solutions (up to a constant)
then implies the global regularity of Aleksandrov solutions.

The optimal transport problem with a general cost yields an equation of the
form (1) with $Y(\cdot,u,Du)=Y(\cdot,Du)$. This equation is more complex.
However thanks to new ideas built on a generalised notion of convexity and a
condition known as A3w, the results of Caffarelli and of Urbas have been
extended to this new equation. Urbas’s results were extended by Trudinger and
Wang [32]. They obtained the estimates for the method of continuity and proved
the existence of globally $C^{3}$ solutions. Again, by the uniqueness up to a
constant, this implies global regularity of Aleksandrov solutions. The
Caffarelli style strict convexity and $C^{1,\alpha}$ regularity under A3w has
been proved by a number of authors, namely Figalli, Kim, and McCann [6],
Guillen and Kitagawa [8], Vètois[34], and Chen and Wang [4]. Each proves the
strict convexity and $C^{1,\alpha}$ regularity under different hypotheses on
the domain and cost function.

For GJEs the picture is not complete. Guillen and Kitagawa [9] have extended
their strict convexity and $C^{1,\alpha}$ regularity results to GJEs. Their
result is of particular interest because their regularity requirements on the
generating function are very weak — it need only be $C^{2}$. However there are
situations where their domain conditions are restrictive. One of which is our
application to global regularity. Thus it’s of interest to have the extension
of Chen and Wang’s result, which holds under weaker conditions on the domains,
to generated Jacobian equations. This extension is one of the goals of this
paper. The strict $g$-convexity implies $C^{1,\alpha}$ regularity, as shown by
Guillen and Kitagawa. We apply the strict convexity result to the problem of
$C^{3}(\overline{\Omega})$ regularity for Aleksandrov solutions. The existence
of $C^{3}(\overline{\Omega})$ solutions was proved by Jiang and Trudinger
[14]. However, Karakhanyan and Wang [15] have shown solutions of generated
Jacobian equations can have different regularity properties. Thus, in stark
contrast to the Monge–Ampère and optimal transport setting, there is no
uniqueness up to a constant and the existence of a globally smooth solution
does not imply the regularity of Aleksandrov solutions. Recently the author
proved if two $C^{1,1}_{\text{loc}}$ solutions intersect then they are the
same solution [26]. We combine this with the strict convexity and Jiang and
Trudinger’s existence result to prove the global regularity of Aleksandrov
solutions as follows: First we introduce a modification of Jiang and
Trudinger’s construction so as to obtain a $C^{3}(\overline{\Omega})$ solution
intersecting a given Aleksandrov solution. By the strict $g$-convexity and
Trudinger’s recent interior regularity result for strictly $g$-convex
solutions [30] we have the required regularity to apply the uniqueness result.
Thus our Aleksandrov solution is the constructed $C^{3}(\overline{\Omega})$
solution.

Here’s the outline of the paper. In Section 2 we introduce $g$-convexity and
the required definitions for our main results. In Section 3 we introduce a
transformation to the generating function and coordinates which makes the
generating function “almost affine”. Such transformations are used in the
optimal transport case for strict convexity [4] and interior regularity [20,
21]. These transformations are the main tool required to extend Chen and
Wang’s strict $c$-convexity; transformations in hand our strict $g$-convexity
result follows a similar framework and uses similar proofs to theirs. We
introduce $g$-cones in Section 4 and estimate their $g$-subdifferentials. In
Section 5 we obtain uniform estimates for Aleksandrov solutions of the
Dirichlet problem for GJEs. The strict $g$-convexity is proved in Section 6
followed by a short proof of $C^{1}$ differentiability in Section 7. We note
at two critical junctures we rely on Lemmas from [6] and [9] and our $g$-cone
and uniform estimates have been obtained under different hypotheses in [9].
Finally in Section 8 we complete the proof of $C^{3}(\overline{\Omega})$
regularity of Aleksandrov solutions as outlined above.

#### Acknowledgements

My thanks to Shibing Chen and Xu-Jia Wang for discussions regarding [4].

##  2\. Generating functions and $g$-convexity

The theory of generated Jacobian equations is a combination of elliptic PDE
and a generalisation of convexity theory. Here we give the definitions
required for the generalised convexity theory. More detailed introductions can
be found in [9, 7, 27]. We begin with the definition of generating functions.
These are a nonlinear extension of affine supporting planes.

######  Definition 1.

A generating function is a function $g$ satisfying the conditions A0,A1,A1∗,
and A2.

A0. The function $g$ satisfies $g\in C^{3}(\overline{\Gamma})$ where $\Gamma$
is a bounded domain of the form $\Gamma:=\\{(x,y,z);x\in U,y\in V,z\in
I_{x,y}\\}\subset\mathbf{R}^{n}\times\mathbf{R}^{n}\times\mathbf{R}$ for
domains $U,V\subset\mathbf{R}^{n}$ and $I_{x,y}$ an open interval for each
$x,y$. Moreover we assume there is an open interval $J$ such that
$g(x,y,I_{x,y})\supset J$ for each $x\in U,y\in V$.

A1.  For each $(x,u,p)\in\mathcal{U}$ defined by

$\mathcal{U}=\\{(x,g(x,y,z),g_{x}(x,y,z));(x,y,z)\in\Gamma\\},$

there is a unique $(x,y,z)\in\Gamma$ such

$\displaystyle g(x,y,z)=u$ $\displaystyle g_{x}(x,y,z)=p.$

A1∗.  For each fixed $y,z$ the mapping $x\mapsto\frac{g_{y}}{g_{z}}(x,y,z)$ is
injective on its domain of definition.

A2. On $\overline{\Gamma}$ there holds $g_{z}<0$ and the matrix111Subscripts
before the comma denote differentiation with respect to $x$, subscripts after
the comma (which are not $z$) denote differentiation with respect to $y$.

$E:=g_{i,j}-g_{z}^{-1}g_{i,z}g_{,j}$

satisfies $\det E\neq 0$.

Later we introduce a dual generating function and we’ll see the A1∗ condition
is simply the A1 condition for the dual generation function — thereby
justifying the name. We’ve incorporated part of Guillen and Kitagwa’s
definition of uniform admissibility condition into A0. The boundedness
requirement on $\Gamma$ may be weakened provided $\|g\|_{C^{3}(\Gamma)}$ is
finite and the quantities in A2 are bounded away from zero.

We define mappings
$Y:\mathcal{U}\rightarrow\mathbf{R^{n}},Z:\mathcal{U}\rightarrow\mathbf{R}$ by
requiring they solve

(5) $\displaystyle g(x,Y(x,u,p),Z(x,u,p))$ $\displaystyle=u,$ (6)
$\displaystyle g_{x}(x,Y(x,u,p),Z(x,u,p))$ $\displaystyle=p.$

We domains $\Omega\subset U$ and $\Omega^{*}\subset V$ and do not forbid
$\Omega=U,\Omega^{*}=V$.

######  Definition 2.

A generated Jacobian equation is an equation of the form (1) where the vector
field $Y$ arises from solving (5),(6) for a generating function.

The basic example is the Monge–Ampère equation which arises from
$g(x,y,z)=x\cdot y-z$. The Monge–Ampère equation is elliptic when solutions
are convex. In this case, that’s when solutions are supported by the
generating function at each point. This permits the following generalisation.

######  Definition 3.

A function $u:\Omega\rightarrow\mathbf{R}$ is called $g$-convex (strictly
$g$-convex) provided for every $x_{0}\in\Omega$ there is $y_{0}\in V$ and
$z_{0}\in\cap_{x\in\Omega}I_{x,y_{0}}$ such that

(7) $\displaystyle u(x_{0})$ $\displaystyle=g(x_{0},y_{0},z_{0})$ (8)
$\displaystyle u(x)$ $\displaystyle\geq(>)g(x,y_{0},z_{0})\text{ for all
}x\in\Omega,x\neq x_{0},$

and for any such $x_{0},y_{0},z_{0}$ we have

(9) $g(\overline{\Omega},y_{0},z_{0})\subset J.$

The containment condition is due to Guillen and Kitagawa [9] and $g$-convex
functions satisfying it are referred to by them as “very nice”. Our definition
of $g$-convex functions implies they are semiconvex.

For $g$-convex $u:\Omega\rightarrow\mathbf{R}$ we define a mapping
$Yu:\Omega\rightarrow V$ as follows

$Yu(x_{0})=\\{y\in V;\text{there is }z_{0}\text{ such that \eqref{eq:conv-
def1}, \eqref{eq:conv-def2} and \eqref{eq:containment} hold}\\}.$

If $u$ is differentiable $Yu(x)=Y(x,u,Du)$. Thus $Yu$ generalises the mapping
$x\mapsto Y(x,u,Du)$ in much the same way the subgradient generalises the
gradient (see Lemma 1 for details). We restrict our attention to $g$-convex
solutions of GJEs. For such solutions the PDE is degenerate elliptic [29].
Using the $Y$ mapping we have the follow generalisation of Aleksandrov
solutions.

######  Definition 4.

Let $u:\Omega\rightarrow\mathbf{R}$ be a $g$-convex function. Then $u$ is
called an Aleksandrov solution of (1) provided for every Borel
$E\subset\Omega$ there holds

$|Yu(E)|=\int_{E}\psi(\cdot,u,Du),$

where $Du$ is well defined almost everywhere by the semiconvexity. When $\psi$
has the form (3) for positive $f,f^{*}$, it is equivalent to require

$\int_{Yu(E)}f^{*}(y)\ dy=\int_{E}f(x)\ dx.$

Given $y_{0}\in Yu(x_{0})$ we frequently need to find the $z_{0}$ for which
$g(\cdot,y_{0},z_{0})$ is a $g$-support at $x_{0}$. This is accomplished by
the dual generating function, which plays several important roles.

######  Definition 5.

The dual generating function is the unique function $g^{*}$-defined on

$\Gamma^{*}:=\\{(x,y,u)=(x,y,g(x,y,z));(x,y,z)\in\Gamma\\},$

by either of the equivalent requirements

(10) $g(x,y,g^{*}(x,y,u))=u\text{ or, equivalently, }g^{*}(x,y,g(x,y,z))=z.$

The dual generating function is well defined because $g_{z}<0$. We note if
$g(\cdot,y_{0},z_{0})$ is a support at $x_{0}$ by (7) we have
$z_{0}=g^{*}(x_{0},y_{0},u(x_{0}))$ and subsequently the support is
$g(\cdot,y_{0},g^{*}(x_{0},y_{0},u(x_{0})))$. Differentiating (10) we obtain
the identities

(11) $\displaystyle g^{*}_{u}=\frac{1}{g_{z}},$ $\displaystyle
g^{*}_{x}=-\frac{g_{x}}{g_{z}},$ $\displaystyle
g^{*}_{y}=-\frac{g_{y}}{g_{z}},$

where $g^{*}$-terms are evaluated at $(x,y,u)$ and $g$-terms at
$(x,y,g^{*}(x,y,u))$, or, alternatively, $g$-terms are evaluated at $(x,y,z)$
and $g^{*}$-terms at $(x,y,g(x,y,z))$.

We also have domain convexity notions.

######  Definition 6.

  1. (1)

A set $A\subset U$ is called (uniformly) $g$-convex with respect to $y\in
V,z\in\cap_{x\in A}I_{x,y}$ provided

$\frac{g_{y}}{g_{z}}(A,y,z),$

is (uniformly) convex.

  2. (2)

A set $B\subset V$ is called (uniformly) $g^{*}$-convex with respect to $x\in
U,u\in J$ provided

$g_{x}(x,\cdot,g^{*}(x,\cdot,u))(B),$

is (uniformly) convex.

Certain statements are made more concise by defining $g$-convexity with
respect to a function, as opposed to points.

######  Definition 7.

Let $u:\Omega\rightarrow\mathbf{R}$ be a $g$-convex function.

  1. (1)

A set $A\subset U$ is called (uniformly) $g$-convex with respect to $u$
provided

$\frac{g_{y}}{g_{z}}(A,y,z),$

is (uniformly) convex whenever $y\in Yu(x),z=g^{*}(x,y,u(x))$ for $x\in\Omega$
($x\in\overline{\Omega}$).

  2. (2)

A set $B\subset V$ is called (uniformly) $g^{*}$-convex with respect to $u$
provided

$g_{x}(x,\cdot,g^{*}(x,\cdot,u(x)))(B),$

is (uniformly) convex for each $x\in\Omega$ ($x\in\overline{\Omega}$).

In line with the above definitions of $g/g^{*}$-convexity, sets whose image
under $x\mapsto\frac{g_{y}}{g_{z}}(x,y,z)$ is a line segment, will be used
repeatedly.

######  Definition 8.

  1. (1)

A collection of points $\\{x_{\theta}\\}_{\theta\in[0,1]}\subset U$ is called
a $g$-segment with respect to $y,z$ provided

$\left\\{\frac{g_{y}}{g_{z}}(x_{\theta},y,z)\right\\}_{\theta\in[0,1]},$

is a line segment.

  2. (2)

A collection of points $\\{y_{\theta}\\}_{\theta\in[0,1]}\subset V$ is called
a $g^{*}$-segment with respect to $x,u$ provided
$\\{g_{x}(x,y_{\theta},g^{*}(x,y_{\theta},u))\\}_{\theta\in[0,1]}$ is a line
segment.

These are the basic definition of $g$-convexity theory. In the optimal
transport case Ma, Trudinger, and Wang [25] introduced a condition on the
fourth derivatives of the cost function to prove interior regularity. Later
work [24, 19, 31, 16, 6] revealed that this condition is essential for the
convexity theory (outlined in Lemma 1). Loeper found a synthetic
interpretation of the condition which has the following extension to
generating functions.

The Loeper Maximum principle (LMP).  Let $x_{0}\in U$, $y_{0},y_{1}\in V$ and
$u_{0}\in J$ be given. Let $\\{y_{\theta}\\}_{\theta\in[0,1]}$ denote the
$g^{*}$-segment with respect $x_{0},u_{0}$ that joins $y_{0}$ to $y_{1}$. The
generating function $g$ satisfies the Loeper maximum principle provided for
all $x\in U$

$g(x,y_{\theta},g^{*}(x_{0},y_{\theta},u_{0}))\leq\text{max}\\{g(x,y_{1},g^{*}(x_{0},y_{1},u_{0})),g(x,y_{0},g^{*}(x_{0},y_{0},u_{0}))\\}.$

The following results, concerning compatibility between the definitions of
$g$-convex functions and $g$-convex sets, are well known consequences of the
Loeper maximum principle [23, 30].

######  Lemma 1.

Assume $g$ is a generating function satisfying the Loeper maximum principle
and $u:\Omega\rightarrow\mathbf{R}$ is a $g$-convex function. Let
$x_{0}\in\Omega,y_{0}\in Yu(x_{0}),u_{0}=u(x_{0})$ and
$z_{0}=g(x_{0},y_{0},u_{0})$. The following statements hold.

  1. (1)

$Yu(x_{0})$ is $g^{*}$-convex with respect to $x_{0},u_{0}$.

  2. (2)

Let $h>0$. Then the sets $\\{x\in\Omega;u(x)<g(x,y_{0},z_{0}-h)\\}$ and
$\\{x\in\Omega;u(x)=g(x,y_{0},z_{0})\\}$ are, when compactly contained in
$\Omega$, $g$-convex with respect to $y_{0},z_{0}-h$, and $y_{0},z_{0}$
respectively.

  3. (3)

$Yu(x_{0})=Y(x_{0},u_{0},\partial u(x_{0}))$ where $\partial u$ denotes the
subdifferential 222The subdifferential is defined for a semiconvex function
$u:\Omega\rightarrow\mathbf{R}$ by  $\partial
u(x_{0})=\\{p\in\mathbf{R}^{n};u(x)\geq
p\cdot(x-x_{0})+u(x_{0})+o(|x-x_{0}|)\text{ for all }x\in\Omega\\}.$ .

Now we have the required terminology to state the conditions for strict
convexity. We assume $\lambda,\Lambda\in\mathbf{R}$ are positive.

######  Theorem 1.

Assume $g$ is a generating function satisfying the Loeper maximum principle.
Assume $\overline{\Omega}\subset U$, and $u:\Omega\rightarrow\mathbf{R}$ is a
$g$-convex function satisfying that for every $E\subset\Omega$

(12) $\lambda|E|\leq|Yu(E)|\leq\Lambda|E|.$

If $U$ and $Yu(\Omega)$ are, respectively, $g$ and $g^{*}$-convex with respect
to $u$, then $u$ is strictly $g$-convex.

It is clear that by redefining $\Gamma$ it suffices there exist any
$U^{\prime}\subset U$ with $\Omega\subset\subset U^{\prime}$ and $U^{\prime}$
$g$-convex with respect to $u$.

We give a short proof of the following consequence of strict $g$-convexity.

######  Corollary 1.

Assume $g$ is a generating function satisfying the Loeper maximum principle.
Assume $u:\Omega\rightarrow\mathbf{R}$ is a strictly $g$-convex solution of
(12). Then $u\in C^{1}(\Omega)$.

Guillen and Kitagawa have proved a stronger conclusion, that strictly
$g$-convex solutions of (12) are in $C^{1,\alpha}(\Omega)$. An obvious
consequence is the $C^{1,\alpha}(\Omega)$ regularity holds under the new
domain hypotheses in Theorem 1.

Our global regularity result uses two additional conditions, called A4w and
A5, on the generating function. We introduce these in Section 8, though note
the A5 condition bounds the gradient of solutions in terms of a particular
constant $K_{0}$. For global regularity the PDE is

(13) $\det DY(\cdot,u,Du)=\frac{f(\cdot)}{f^{*}(Y(\cdot,u,Du))}\quad\text{in
}\Omega,$

for positive densities $f,f^{*}$ and we assume the second boundary value
problem (4) is satisfied. Then a necessary condition for the existence of a
$C^{2}(\Omega)$ $g$-convex solution is the mass balance condition

(14) $\int_{\Omega}f=\int_{\Omega^{*}}f^{*}.$

######  Theorem 2.

Let $\overline{\Omega}\subset U$, $\overline{\Omega^{*}}\subset V$ be $C^{4}$
domains, $g$ a $C^{4}$ generating function satisfying LMP, A4w, A5, and,
finally, $f\in C^{2}(\overline{\Omega}),f^{*}\in C^{2}(\overline{\Omega^{*}})$
be positive functions satisfying (14). Suppose $u\in C^{0}(\overline{\Omega})$
is a $g$-convex Aleksandrov solution of (13),(4) satisfying the following
property: there is $x_{0}\in\Omega$ and a support $g(\cdot,y_{0},z_{0})$ at
$x_{0}$ such that
$J_{0}:=[\inf_{\Omega}g(\cdot,y_{0},z_{0})-K_{0}\text{diam}(\Omega),\sup_{\Omega}g(\cdot,y_{0},z_{0})+K_{0}\text{diam}(\Omega)]\subset
J$. Assume $\Omega^{*}$ is uniformly-$g^{*}$-convex with respect to points in
$\Omega\times J_{0}$ and $\Omega$ is uniformly-$g$-convex with respect to
points in $\Omega^{*}\times g^{*}(\Omega,\Omega^{*},J_{0})$. Then $u\in
C^{3}(\overline{\Omega})$.

##  3\. Transformations

In this section we introduce transformations which leave the generating
function close to $g(x,y,z)=x\cdot y-z$. Assume $x_{0}\in U,y_{0}\in
V,u_{0}\in J$ and $h\geq 0$ are given. For context, we usually have a
$g$-convex function $u:\Omega\rightarrow\mathbf{R}$ and take $x_{0}\in\Omega$
with $y_{0}\in Yu(x_{0}),$ $u_{0}=u(x_{0})$ and shift the support to
$g(\cdot,y_{0},g^{*}(x_{0},y_{0},u_{0}+h))$. Without loss of generality
$x_{0},y_{0},u_{0}=0$. Set $z_{h}=g^{*}(0,0,h)$. After replacing $g$ by the
function $(x,y,z)\mapsto g(x,y,z+g^{*}(0,0,h))$ we assume $g(0,0,0)=h$ so
$g^{*}(0,0,h)=0$. Furthermore by working in the coordinates
$y^{\prime}:=E(0,0,0)y$ we have $E(0,0,0)=\text{Id}$. (We recall $E$ is the
matrix from A2.)

### Transformed coordinates

Define

(15) $\displaystyle q(x)$
$\displaystyle:=g_{z}(0,0,0)\left[\frac{g_{y}}{g_{z}}(x,0,0)-\frac{g_{y}}{g_{z}}(0,0,0)\right],$
(16) $\displaystyle p(y)$
$\displaystyle:=g_{x}(0,y,g^{*}(0,y,h))-g_{x}(0,0,0).$

Conditions A1,A1∗, and A2 imply $x\mapsto q(x)$ and $y\mapsto p(y)$ are
diffeomorphisms, so we may write $q=q(x)$, or $x=x(q)$ as necessary, similarly
for $y$ and $p$. The Jacobian of the first transform is

(17) $\frac{\partial q_{i}}{\partial
x_{j}}=\frac{g_{z}(0,0,0)}{g_{z}(x,0,0)}E_{ji}(x,0,0).$

Because $g_{z}\det E\neq 0$ on $\overline{\Gamma}$ these transformations are
non-degenerate. It is useful to introduce a quantity which quantifies this
nondegeneracy and how far the generating function is from $g(x,y,z)=x\cdot
y-z$. Put

(18) $\displaystyle
E^{+}=\text{sup}_{\Gamma}\sup_{\xi\in\mathbf{S}^{n-1}}|E_{ij}\xi_{i}|$
$\displaystyle
E^{-}=\text{inf}_{\Gamma}\inf_{\xi\in\mathbf{S}^{n-1}}|E_{ij}\xi_{i}|,$ (19)
$\displaystyle C_{z}=\sup_{\Gamma}|g_{z}|$ $\displaystyle
c_{z}=\inf_{\Gamma}|g_{z}|,$

and set $C_{g}=\max\\{E^{+},C_{z},1/E^{-},1/c_{z}\\}$, where if
$g(x,y,z)=x\cdot y-z$ we have $C_{g}=1$. $C_{g}$ is used to quantify the
effect of $g$ not being affine. For example if $D\subset U$ and $D_{q}$ is its
image under (15), then by (17) there holds

(20) $C_{g}^{-3}|D|\leq|D_{q}|\leq C_{g}^{3}|D|.$

A similar estimate, depending only on $C_{g}$, holds for the $y$ to $p$
transformation.

### Generating function transformation

Set

$\tilde{g}(x,y,z)=\frac{g_{z}(0,0,0)}{g_{z}(x,0,0)}[g(x,y,g^{*}(0,y,h-z))-g(x,0,0)],$

and subsequently

$\overline{g}(q,p,z)=\tilde{g}(x(q),y(p),z),$

where $x,q$ and $y,p$ satisfy (15) and (16) respectively. As motivation note
in the optimal transport case, where $g(x,y,z)=c(x,y)-z$ for $c$ the cost
function, we have

$\tilde{g}(x,y,z)=[c(x,y)-c(0,y)]-[c(x,0)-c(0,0)]-z,$

which is a frequently used transformation [6, 21, 4]. We note Jhaveri [12] has
made use of a different transformed generating function.

The basic facts concerning $\overline{g}$ are summarized in the following
lemma. We use the overline notation to denote quantities corresponding to
$\overline{g}$.

######  Lemma 2.

Let $g$ be a generating function satisfying the LMP. Then:
(1) $\overline{g}$ is a $C^{2}$ generating function satisfying the LMP.
(2) A function $u$ is $g$-convex if and only if the corresponding function

(21)
$\overline{u}(q):=\frac{g_{z}(0,0,0)}{g_{z}(x(q),0,0)}[u(x(q))-g(x(q),0,0)],$

is $\overline{g}$-convex. Moreover, with $\overline{Y}$ defined for
$\overline{g}$ as $Y$ was for $g$, we have $y\in Yu(x)$ if and only if
$p(y)\in\overline{Y}\overline{u}(q(x))$.

The proof is a direct calculation which we defer to Appendix A. Again, the
effect of these transformations can be measured in terms of $C_{g}$. In
particular for $\overline{u}$ as in Lemma 2

(22) $C_{g}^{-2}|\overline{u}(q)|\leq|u(x(q))-g(x(q),0,0)|\leq
C_{g}^{2}|\overline{u}(q)|$

We emphasize this because we will frequently consider $\overline{u}(q)$ in
place of $u(x(q))-g(x(q),0,0)$ in certain estimates. The estimate (22) implies
this is acceptable as these quantities are comparable up to the constant
$C_{g}$. Near the origin $\overline{g}$ is close to a plane in the sense of
the following lemma. Such expansions are important for the regularity in
optimal transport [20, 21].

######  Lemma 3.

Let $g$ be a generating function and $\overline{g}$ be the transformed
generating function. There are $C^{1}$ functions
$a^{(\alpha)}(q,p),b^{(\beta)}(q,p),f^{(\gamma)}(q,p,z)$ for
$\alpha,\beta=1,2,3$ and $\gamma=1,2$, which arise as Taylor series remainder
terms, such that $\overline{g}$ satisfies the following identities

(23) $\displaystyle\overline{g}(q,p,z)$ $\displaystyle=q\cdot
p-z+a^{(1)}_{ijk}(q,p)q_{i}q_{j}p_{k}+f^{(1)}(q,p,z)z,$ (24)
$\displaystyle\overline{g}(q,p,z)$ $\displaystyle=q\cdot
p-z+a^{(2)}_{ijk}(q,p)q_{i}p_{j}p_{k}+f^{(1)}(q,p,z)z.$

Here $f^{(1)}$ satisfies the inequalities

(25) $\displaystyle-C^{-}\leq-1+f^{(1)}(q,p,z)$ $\displaystyle\leq-C^{+},$
(26) $\displaystyle\text{ and }|f^{(1)}(q,p,z)|$ $\displaystyle\leq C|q|$

for positive constants $C^{\pm}$ depending only on $C_{g}$ and $C$ depending,
in addition, on $sup|g_{xz}|$. Furthermore $f^{(1)}$ satisfies the equality

(27) $\displaystyle
f^{(1)}(q,p,z)=b^{(1)}_{ij}q_{i}q_{j}+b^{(2)}_{ij}q_{i}p_{j}+b^{(3)}_{ij}p_{i}p_{j}+f^{(2)}(q,p,z)z.$

###### Proof.

First, write

(28)
$\displaystyle\overline{g}(q,p,z)=\overline{g}(q,p,0)+\int_{0}^{1}\overline{g}_{z}(q,p,tz)z\
dt.$

We take $f^{(1)}(q,p,z)=(1+\int_{0}^{1}\overline{g}_{z}(q,p,tz)\ dt)z$. Then
$f^{(1)}$ satisfies (25) because by a direct calculation
$-C_{g}^{4}\leq\overline{g}_{z}\leq-C_{g}^{-4}$. Note here, and throughout
this proof, we rely on the calculations (11). Next, because
$\overline{g}_{z}(0,p,z)=-1$, (26) is the Lipschitz continuity of this
quantity, as guaranteed by A0. To obtain (27) we expand with a Taylor series
in $z$, then in $q,p$ and obtain

(29) $\displaystyle\overline{g}_{z}$
$\displaystyle(q,p,z)=\overline{g}_{z}(q,p,0)z+\frac{1}{2}\overline{g}_{zz}(q,p,\tau
z)z^{2}$ (30)
$\displaystyle=\overline{g}_{z}(0,0,0)z+\overline{g}_{q_{i},z}(0,0,0)q_{i}z+\overline{g}_{p_{i},z}(0,0,0)p_{i}z+g_{zz}(q,p,\tau
z)z^{2}$
$\displaystyle\quad+z[\overline{g}_{q_{i},p_{j},z}(q_{t},p_{t},0)q_{i}p_{j}+\overline{g}_{q_{i}q_{j},z}(q_{t},p_{t},0)q_{i}q_{j}+\overline{g}_{p_{i}p_{j},z}(q_{t},p_{t},0)p_{i}p_{j}],$

where $t,\tau\in(0,1)$, $q_{t}=tq$ and similarly for $p_{t}$. Using
$\overline{g}_{z}(0,p,0)=\overline{g}_{z}(q,0,0)=-1,$ and subsequently
$\overline{g}_{q_{i},z}(0,0,0)=\overline{g}_{p_{i},z}(0,0,0)=0$, (30) implies
(27). Whilst we’ve not explicitly used it, the integral form of the remainder
term implies the coefficients of $q_{i}q_{j},q_{i}p_{j},p_{i}p_{j}$ are
$C^{1}$.

We’re left to deal with the term $\overline{g}(q,p,0)$ in (28). Set
$\tilde{c}(x,y):=\tilde{g}(x,y,0)$ and $c(q,p)=\tilde{c}(x(q),y(p))$. This
suggestive notation indicates all the following calculations are based on the
optimal transport case [6, 4, 21]. By direct computation $\tilde{c}$ satisfies
$\tilde{c}_{x}(0,y)=p$ and $\tilde{c}_{y}(x,0)=q$ along with
$\tilde{c}(x,0)\equiv 0,$ $\tilde{c}(0,y)\equiv 0$ and
$\tilde{c}_{i,j}(0,0)=\delta_{ij}$. It’s these identities that justify our
inclusion of the $g_{z}$ terms in the definitions of $q,p,\overline{g}$. They
imply

(31) $\displaystyle c(0,0)=0$ $\displaystyle c_{q}(0,0)=0$ $\displaystyle
c_{p}(0,0)=0$ (32) $\displaystyle c_{p_{i},q_{j}}(0,0)=\delta_{ij}$
$\displaystyle c_{qq,p}(q,0)=0$ $\displaystyle c_{q,pp}(0,p)=0.$

Thus via a Taylor series

(33) $\displaystyle c(q,p)$
$\displaystyle=c(0,p)+c_{q_{i}}(0,p)q_{i}+c_{q_{i},q_{j}}(tq,p)q_{i}q_{j},$

As usual $t,\tau\in(0,1)$ (we’re about to use $\tau$). More Taylor series and
we obtain

(34) $\displaystyle c_{q_{i}}(0,p)$
$\displaystyle=c_{q_{i}}(0,0)+c_{q_{i},p_{j}}(0,0)p_{j}+c_{q_{i},p_{j}p_{k}}(0,\tau
p)p_{j}p_{k}=p_{i}$ (35) $\displaystyle c_{q_{i}q_{j}}(tq,p)$
$\displaystyle=c_{q_{i}q_{j}}(tq,0)+c_{q_{i}q_{j},p_{k}}(tq,\tau
p)p_{k}=a_{ijk}^{(1)}p_{k}.$

The integral form of the remainder term implies $a^{(1)}$ is $C^{1}$.
Combining (33)-(35) yields (23). Similar calculations imply (24). ∎

######  Remark 1.

When $g$ is $C^{4}$ an additional term in the Taylor series (35) yields

(36) $\displaystyle\overline{g}(q,p,z)$ $\displaystyle=q\cdot
p-z+a^{(3)}_{ij,kl}(q,p)q_{i}q_{j}p_{k}p_{l}+f^{(1)}(q,p,z)z,$

though we don’t use this expansion here.

##  4\. $g$-cones

Cones are a basic tool for studying the Monge–Ampère equation. A similar class
of functions was introduced in the optimal transport setting by Figalli, Kim,
and McCann [6, Section 6.2]. The defining feature of this so-called $c$-cone
is that its $Y$ mapping is concentrated at a point. The generalization to
$g$-cones is due to Guillen and Kitagawa [9]. In each case we want estimates
for the $Y$-mapping of the generalised cone in terms of the base and height of
the generalised cone.

Let $u:\Omega\rightarrow\mathbf{R}$ be a $g$-convex function. Assume
$x_{0}\in\Omega,y_{0}\in V$ are given and $u_{0}:=u(x_{0})$. For $h>0$ small
set $z_{h}=g^{*}(x_{0},y_{0},u_{0}+h)$ and assume

(37) $D:=\\{x\in\Omega;u(x)<g(x,y_{0},z_{h})\\}\subset\subset\Omega.$

We define the $g$-cone with vertex $(x_{0},u_{0})$ and base
$\\{(x,g(x,y_{0},z_{h}));x\in\partial D\\}$ by

$\displaystyle\vee(x)$
$\displaystyle=\sup\\{\varphi_{y}(x):=g(x,y,g^{*}(x_{0},y,u_{0}));\varphi_{y}(x)\leq
g(x,y_{0},z_{h})\text{ on }\partial D\\}.$

When we need to emphasize dependencies we include them as a subscript, e.g.
$\vee_{D,h}$ if $x_{0},y_{0},u_{0}$ are clear from context. The expression for
$\vee$ is well defined for any $D\subset U$. However requiring $D$ satisfy
(37) ensures $\vee=g(\cdot,y_{0},z_{h})$ on $\partial D$.

Our goal is to estimate $Y\vee_{D,h}(x_{0})$ in terms of $D$ and $h$. As in
Section 3 we assume, without loss of generality, that
$x_{0},y_{0},u_{0},z_{h}=0$. Using Lemma 2(2) it suffices to work in the
coordinates given by (15), (16) and estimate the $\overline{Y}$ mapping of

$\overline{\vee}(q):=\frac{g_{z}(0,0,0)}{g_{z}(x(q),0,0)}[\vee(x(q))-g(x(q),0,0)].$

By direct calculation we see $\overline{\vee}$ is the $\overline{g}$-cone with
base $\partial D_{q}\times\\{0\\}$ and vertex $(0,-h)$ (recall $D_{q}$ is the
image of $D$ in the $q$ coordinates). Thus

(38)
$\overline{\vee}(q)=\sup\\{\varphi_{p}(q):=\overline{g}(q,p,h);\varphi_{p}\leq
0\text{ on }D_{q}\\},$

with $D_{q}$ convex and $\overline{g}$ satisfying Lemma 3. To simplify
notation we revert to $x,y,g,\vee$.

We compare $\vee$ with a cone of the same base and height. The cone with
vertex $(0,-h)$ and base $\partial D\times\\{0\\}$ is
$K:D\rightarrow\mathbf{R}$ defined by

(39) $K(x)=\sup\\{l_{p}(x):=p\cdot x-h;p\in\mathbf{R}^{n}\text{ and }l_{p}\leq
0\text{ on }\partial D\\}.$

###  4.1. Upper bounds for $Y\vee(0)$

######  Lemma 4.

Suppose $g$ is a generating function satisfying LMP and the identities in
Lemma 3. Let $D$ be a convex domain containing 0. Suppose $\vee,K$ are as
defined in (38),(39) respectively. There is $d_{0},h_{0}>0$ such that if
$\text{diam}(D)\leq d_{0}$ and $h\leq h_{0}$ then

(40) $Y\vee(0)\subset 2\partial K(0).$

Where $d_{0},h_{0}$ depend on $\|g\|_{C^{3}}$ and $2\partial
K(0)=\partial(2K)(0)$.

###### Proof.

We prove the transformed generating function satisfies

(41) $g(x,y,h)\geq\frac{3}{4}x\cdot y-\frac{3h}{2},$

for $|x|,h$ sufficiently small and $x\cdot y,h>0$,333we note if $x\cdot y$ or
$h<0$ (41) holds with $3/4$ replaced by $5/4$ or $3/2$ replaced by $1/2$. ;
(40) is a straightforward consequence. Indeed, take $y\in Y\vee(0)$ and
suppose $y\notin 2\partial K(0)$, that is $x\cdot y>2h$ for some boundary
point $x\in\partial D$. By (41) $g(x,y,h)>0$ and so $g(\cdot,y,h)$ can not be
a support of $\vee$.

Take $y\in V$ and rotate so $y=(0,\dots,0,y_{n})$. Let
$x=(x_{1},\dots,x_{n})\in D$ satisfy $x_{n}y_{n}>0$ and set
$x^{\prime}=(x_{1},\dots,x_{n-1},0)$. We assume, for now,
$g(x^{\prime},y,0)\geq 0$ (we’ll see this is a consequence of LMP). Now, (23)
implies

(42) $g_{x_{n}}(x_{\tau},y,0)x_{n}\geq x_{n}y_{n}-K|x|x_{n}y_{n},$

for $x_{\tau}=\tau x+(1-\tau)x^{\prime}$ and $\tau\in[0,1]$ where $K$ depends
on $\|g\|_{C^{3}}$. By (26) and a choice of $\text{diam}(D)$ small

$\displaystyle g(x,y,h)\geq g(x,y,0)-\frac{3h}{2}.$

A Taylor series for $h(t):=g(tx+(1-t)x^{\prime},y,0)$, our assumption
$g(x^{\prime},y,0)\geq 0$, and (42) imply

$\displaystyle g(x,y,h)$ $\displaystyle\geq
g(x^{\prime},y,0)+g_{x_{n}}(x_{\tau},y,0)x_{n}-\frac{3}{2}h\geq
x_{n}y_{n}(1-K|x|)-\frac{3}{2}h.$

Choosing $\text{diam}(D)$ small to ensure $K|x|\leq 1/4$ we obtain (41).

To conclude we show $g(x^{\prime},y,0)\geq 0$. Since $x^{\prime}\cdot y=0$ it
suffices to show whenever $x\cdot y>0$ then $g(x,y,0)\geq 0$ and use
continuity. Note if $x\cdot y>0$ the expression (23) implies

$g(tx,y,0)>0\text{ and }g(-tx,y,0)<0,$

for $t>0$ sufficiently small. If $g(x,y,0)<0$ then the $g$-convexity of the
section $\\{g(\cdot,y,0)<0=g(\cdot,0,0)\\}$ with respect to $0,0$ (which is
just convexity), is violated. So as required $g(x,y,0)\geq 0$. ∎

###  4.2. Lower bounds for $\partial\vee(0)$

The estimates in the other direction are formulated differently. As motivation
consider the rectangle

(43) $R=\\{x\in\mathbf{R}^{n};-b_{i}\leq x_{i}\leq a_{i}\\},$

for $a_{i},b_{i}>0$, and the cone $K$ with base $R$ and vertex $(0,-h)$. Then
$\partial K(0)$ contains the points $he_{i}/a_{i},-he_{i}/b_{i}$. Thus with

(44) $R^{*}:=\\{x\in\mathbf{R}^{n};-b_{i}^{-1}\leq x_{i}\leq a_{i}^{-1}\\}.$

we have

(45) $\displaystyle\partial K(0)$ $\displaystyle\supset C_{n}hR^{*}$ (46)
$\displaystyle|\partial K(0)|$ $\displaystyle\geq
C_{n}h^{n}\prod_{i=1}^{n}\left(\frac{1}{b_{i}}+\frac{1}{a_{i}}\right).$

Next we decrease the base of the cone: consider a domain $D$ with $0\in
D\subset R$ and the cone $K_{D}$ with base $D$ and vertex $(0,-h)$. Because
$\partial K(0)\subset\partial K_{D}(0)$, (45) and (46) hold with $K$ replaced
by $K_{D}$. This motivates the following result.

######  Lemma 5.

Suppose $g$ is a generating function satisfying LMP and the identities in
Lemma 3. Let $D$ be a convex domain with $0\in D\subset R$. Let $\vee$ be
given by (38). There is $d_{0},h_{0}>0$ such that if $\text{diam}(D)\leq
d_{0}$ and $h\leq h_{0}$ then (45) and (46) hold with $\vee$ in place of $K$.
The quantities $d_{0},h_{0}$ depend on $\text{diam}(V),\|g\|_{C^{3}}$,

###### Proof. (Lemma 5).

Rather than showing $\partial\vee(0)$ contains the points $Che_{i}/a_{i}$ and
$-Che_{i}/b_{i}$ for $i=1,\dots,n$ we will, instead, show $\partial{\vee}(0)$
contains points close to these points. That is, we show for for some
$\kappa\geq 1/4$ and $q:=\kappa he_{n}/a_{n}$ there is $p\in\partial\vee(0)$
satisfying

(47) $|p-q|\leq\frac{1}{16}|q|.$

Our proof also applies to $\kappa he_{i}/a_{i}$ and $-\kappa he_{i}/b_{i}$ for
$i=1,\dots,n$, so $ChR^{*}\subset\partial\vee(0)$.

To begin, choose $\hat{x}$ realizing
$\hat{x}_{n}=\sup\\{x_{n};x=(x_{1},\dots,x_{n})\in D\\}$. We see, by taking a
limit of the $\varphi_{y}$ used in (38) to define $\vee$, that there is
$\hat{y}$ for which $g(\cdot,\hat{y},h)$ supports $\vee$ at $\hat{x}$ and $0$.
In particular, since $\vee=0$ at $\hat{x}$ and is less than or equal to $0$ on
$\partial D$ we have, for $\hat{y}$ appropriately chosen and some $\beta\geq
0$,

(48) $g_{x}(\hat{x},\hat{y},h)=\beta e_{n}.$

We’ll prove that $p=g_{x}(0,\hat{y},h)$ and $q=(p\cdot e_{n})e_{n}$ satisfy
(47).

Choose $d^{*}$ so that $g(d^{*}e_{n},\hat{y},h)=0$. We claim $d^{*}\leq
a_{n}$. Indeed

$S:=\\{x;g(x,\hat{y},h)<0=g(x,0,0)\\}$

is $g$-convex with respect to $(0,0)$, that is, convex. Furthermore since
$g_{x}(\hat{x},\hat{y},h)=\beta e_{n}$ and $g(\hat{x},\hat{y},h)=0$, the plane
$P:=\\{x;x_{n}=\hat{x}_{n}\\}$ supports $S$. Thus, since $S$ contains $0$ and
lies on one side of $P$, $S\subset\\{x;x_{n}\leq\hat{x}_{n}\\}$ and $d^{*}\leq
a_{n}$.

Now (23) with (27) implies

(49) $|g_{x}(x,\hat{y},h)-g_{x}(0,\hat{y},h)|\leq
C(|x|+h)||\hat{y}|+Kh(h+|x|),$

where $C,K$ depend on $\|g\|_{C^{3}},$ $\text{diam}(V)$ and we now assume
$\text{diam}(D)\leq 1$. By a Taylor series we obtain for some $\tau\in(0,1)$

$\displaystyle h$ $\displaystyle=g(d^{*}e_{n},\hat{y},h)-g(0,\hat{y},h)$
$\displaystyle=d^{*}g_{x_{n}}(\tau d^{*}e_{n},\hat{y},h)$ (50)
$\displaystyle\leq
d^{*}|g_{x}(0,\hat{y},h)|+Cd^{*}(d^{*}+h)|\hat{y}|+Kh(h+d^{*}).$

To estimate $|\hat{y}|$ in terms of $|g_{x}(0,\hat{y},h)|$ write

(51)
$|\hat{y}|=|g_{x}(0,\hat{y},0)|\leq|g_{x}(0,\hat{y},h)|+|g_{xz}(0,\hat{y},\tau
h)|h.$

Combining (50) and (51) we have

$h\leq d^{*}|g_{x}(0,\hat{y},h)|[1+Cd^{*}(d^{*}+h)]+Kh(h+d^{*}).$

We choose $\text{diam}(D)$ and $h$ small to ensure both
$(1+Cd^{*}(d^{*}+h))\leq 3/2$ and $K(h+d^{*})\leq 1/4$. Combining with
$d^{*}\leq a_{n}$ yields

(52) $\frac{h}{2a_{n}}\leq|g_{x}(0,\hat{y},h)|.$

Using, once again, (49) (this time with $x=\hat{x}$) and (51) we have

$|g_{x}(\hat{x},\hat{y},h)-g_{x}(0,\hat{y},h)|\leq
C(|\hat{x}|+h)||g_{x}(0,\hat{y},h)|+Kh(|\hat{x}|+h).$

Dividing through by $|g_{x}(0,\hat{y},h)|$, using (52) and choosing
$h,|\hat{x}|$ sufficiently small we can ensure

$\left|\frac{g_{x}(\hat{x},\hat{y},h)}{|g_{x}(0,\hat{y},h)|}-\frac{g_{x}(0,\hat{y},h)}{|g_{x}(0,\hat{y},h)|}\right|\leq
1/16.$

The first vector lies on the $e_{n}$ axis (recall (48)). Thus the unit vector
$\frac{g_{x}(0,\hat{y},h)}{|g_{x}(0,\hat{y},h)|}$, and consequently
$g_{x}(0,\hat{y},h)$ make angle $\theta$ with the $e_{n}$ axis for $\theta$
satisfying $\sin(\theta)\leq 1/16$, i.e. $\theta\leq 1/8$. This, with (52)
implies both

$e_{n}\cdot
g_{x}(0,\hat{y},h)=\cos(\theta)|g_{x}(0,\hat{y},h)|\geq\frac{h}{4a_{n}},$

and

$|g_{x}(0,\hat{y},h)-(e_{n}\cdot
g_{x}(0,\hat{y},h))e_{n}|\leq\sin(\theta)|g_{x}(0,\hat{y},h)|\leq\frac{1}{16}|(e_{n}\cdot
g_{x}(0,\hat{y},h))e_{n}|,$

which is (47). ∎

We have a more precise estimate when $x_{0}$ is close to the boundary. We make
use of the minimum ellipsoid (see [22, Section 2.1]) and the following
(specific case of a) lemma due to Figalli, Kim, and McCann [6].

######  Lemma 6.

[6, Lemma 6.9] Let $D\subset\mathbf{R}^{n}$ be a convex domain. Assume $D$
contains a “vertical” line segment
$\\{(x^{\prime},t_{0}+t);x^{\prime}\in\mathbf{R}^{n-1},t\in[0,d]\\}$ of length
$d$. Let

$D^{\prime}:=\\{(x_{1},\dots,x_{n-1},0);x=(x_{1},\dots,x_{n})\in D\\}$

be the projection of $D$ onto $\mathbf{R}^{n-1}$. There is $C>0$ depending
only on $n$ such that

$|D|\geq Cd\mathcal{H}^{n-1}(D^{\prime}),$

where $\mathcal{H}^{n-1}$ is the $n-1$ dimensional Hausdorff measure.

Lemma in hand, we prove the estimate close to the boundary using a similar
proof to that of Figalli, Kim, and McCann.

######  Lemma 7.

Suppose $g$ is a generating function satisfying LMP and the identities in
Lemma 3. Let $D$ be a convex domain with $0\in D$. Let $\vee$ be given by
(38). Assume $0$ is close to the boundary, in the sense that there is a unit
vector $\nu$ and positive $d$ such that

(53) $\sup_{x\in D}\langle x,\nu\rangle=\varepsilon d,$

and in addition $D$ contains a line segment of length $d$ parallel to $\nu$.
There is $C,d_{0},h_{0}>0$ depending on $\|g\|_{C^{3}}$, such that if
$\text{diam}(D)\leq d_{0}$ and $h\leq h_{0}$ then

$h^{n}\leq C\varepsilon|\partial\vee(0)||D|.$

###### Proof.

We assume, without loss of generality that $\nu=e_{n}$. Let $D^{\prime}$ be as
in Lemma 6. Then up to a choice of the remaining coordinates we assume the
minimum ellipsoid of $D^{\prime}$ (as a subset of $\mathbf{R}^{n-1}$) is

$E^{\prime}:=\\{x^{\prime}=(x_{1},\dots,x_{n-1},0);\sum_{i=1}^{n-1}\left(\frac{x_{i}-\overline{x}_{i}}{b_{i}/2}\right)^{2}\leq
1\\},$

for some $\overline{x}\in D^{\prime}$. Then $\mathcal{H}^{n-1}(D^{\prime})\geq
C_{n}b_{1}\dots b_{n-1}$ and

$D\subset[-b_{1},b_{1}]\times\dots\times[-b_{n-1},b_{n-1}]\times[-K,\varepsilon
d],$

for some $K>0$. Then Lemma 5, specifically (46), implies

(54) $|\partial\vee(0)|\geq C_{n}h^{n}\frac{1}{\varepsilon db_{1}\dots
b_{n-1}}.$

On the other hand Lemma 6 implies $|D|\geq C_{n}d|D^{\prime}|\geq
C_{n}db_{1}\dots b_{n-1}$, which combined with (54) completes the proof. ∎

##  5\. Uniform estimates

In this section we consider $g$-convex Aleksandrov solutions of

(55) $\displaystyle\lambda$ $\displaystyle\leq\det DYu\leq\Lambda\text{ in
}D,$ (56) $\displaystyle u$ $\displaystyle=g(\cdot,y_{0},z_{0})\text{ on
}\partial D.$

Here $\lambda,\Lambda$ are positive constants and $D$ (being a section) is
necessarily $g$-convex with respect to $y_{0},z_{0}$. Our goal is to estimate
$|u-g(\cdot,y_{0},z_{0})|$ in D. By using the $g$-cone estimates our proofs
are direct extensions of those in the Monge–Ampère case. In this section we
assume $\text{diam}(D)$ and $h:=\sup|u-g(\cdot,y_{0},z_{0})|$ are sufficiently
small as required by Lemmas 4, 5 and 7.

######  Theorem 3.

Assume $g$ is a generating function satisfying LMP. Assume $u$ is a $g$-convex
solution of (55), (56). There is $C>0$ depending only on $\Lambda,n,C_{g}$
such that

(57) $\sup_{D}|u(\cdot)-g(\cdot,y_{0},z_{0})|^{n}\leq C|D|^{2}.$

###### Proof.

Fix any $x_{0}\in D$, after translating assumed to be 0. It suffices to obtain
(57) after applying the transformations in Section 3. Estimate (57) will hold
for the original function and coordinates with appropriate inclusion of the
constant $C_{g}$. Thus we assume $D$ is convex, $g$ satisfies the identities
in Lemma 3 and $u\equiv 0$ on $\partial D$. Let the minimum ellipsoid of $D$
be given by

$E:=\big{\\{}x;\sum\frac{(x_{i}-\overline{x}_{i})}{a_{i}^{2}}\leq 1\big{\\}},$

for some $\overline{x}\in D$. Note $D\subset R:=\\{-2a_{i}\leq x_{i}\leq
2a_{i}\\}$. Set $R^{*}=\\{-(2a_{i})^{-1}\leq x_{i}\leq(2a_{i})^{-1}\\}$ and
note $|R|^{*}\geq C_{n}|D|^{-1}$. Thus by Lemma 5 the $g$-cone $\vee$ with
vertex $(0,u(0))$ and base $\partial D\times\\{0\\}$ satisfies
$C|D||\partial\vee(0)|\geq|u(0)|^{n}$. Furthermore
$Y\vee(0)=Y(0,\vee(0),\partial\vee(0))$ so $|Y\vee(0)|\geq C|\partial\vee(0)|$
for $C$ depending on $C_{g}$. Thus

(58) $|D||Y\vee(0)|\geq C|u(0)|^{n}.$

The comparison principle ([29, Lemma 4.4]) implies $Y\vee(0)\subset Yu(D)$ so
(58) along with $|Yu(D)|\leq\Lambda|D|$ implies the result. ∎

If instead we use Lemma 7 we obtain the following:

######  Theorem 4.

Assume $g$ satisfies LMP and the identities in Lemma 3 and that $D$ is a
convex domain. Let $u$ satisfy (55) and (56). Suppose further that $x_{0}\in
D$ is a point close to the boundary in the sense that there is a unit vector
$\nu$ and $\varepsilon,d>0$ with

$\sup_{x\in D}\langle x-x_{0},\nu\rangle=\varepsilon d$

and $D$ contains a line segment of length $d$ parallel to $\nu$. There is
$C>0$ independent of $u$ such that

$|u(x_{0})-g(x_{0},y_{0},z_{0})|^{n}\leq C\varepsilon|D|^{2}.$

######  Remark 2.

For Theorems 3 and 4 it suffices that $u\geq g(\cdot,y_{0},z_{0})$ on
$\partial D$. In this case we apply the above proofs to $D^{\prime}=\\{u\leq
g(\cdot,y_{0},z_{0})\\}\subset D$ which is assumed nonempty. Similarly for the
lower bound, Theorem 5, it suffices to have $u\leq g(\cdot,y_{0},z_{0})$ on
$\partial D$.

The lower bound uses a lemma of Guillen and Kitagawa’s [9]. We use it only in
the case of the transformed generating function. This simplifies the proof,
which we give after the proof of Theorem 5.

######  Lemma 8.

[9, Lemma 6.1] Let $g$ be the transformed generating function from Lemma 2.
Suppose $u$ is $g$-convex and $D:=\\{u\leq 0\\}$ has $0$ as the centre of its
minimum ellipsoid. Set $h=\sup_{D}|u|$. There are constants $C,K>0$ depending
only $\|g\|_{C^{3}}$ such that

$Yu\left(\frac{1}{K}D\right)\subset Y\vee_{D,Ch}(0),$

where $\vee_{D,Ch}$ is the $g$-cone with base $D\times\\{0\\}$ and vertex
$(0,-Ch)$.

######  Theorem 5.

Assume $g$ is a generating function satisfying LMP and $u$ is a $g$-convex
solution of (55),(56). There is a constant $C$ depending on
$\lambda,n,\|g\|_{C^{3}}$ such that

$C|D|^{2}\leq\sup_{D}|u(\cdot)-g(\cdot,y_{0},z_{0})|^{n}.$

###### Proof.

We assume $y_{0},z_{0}=0$ then pick any $x_{0}\in D$ and make the change (15)
so that $D$ is convex. After translation we assume the minimum ellipsoid for
$D$ is centred on $0$. We make the remaining changes in Section 3 so $g$
satisfies Lemma 3 and $u=0$ on $\partial D$.

Let $\vee=\vee_{\frac{1}{K}D,Ch}$ be the $g$-cone from Lemma 8 and note

$\frac{\lambda}{K^{n}}|D|\leq\left|Yu\left(\frac{1}{K}D\right)\right|\leq|Y\vee(0)|.$

Let $\hat{K}$ be the classical cone with the same base and vertex as $\vee$.
By Lemma 4

(59) $\frac{\lambda}{K^{n}}|D|\leq C|\partial\hat{K}(0)|.$

The estimate

(60) $|\partial\hat{K}(0)|\leq\frac{Ch^{n}}{|D|},$

follows from standard convex geometry making use of the minimum ellipsoid.
Combining (59) and (60) completes the proof. ∎

The proof of Lemma 8 uses the Loeper maximum principle via the quantitative
quasiconvexity interpretation of Guillen and Kitagawa. More precisely if the
Loeper maximum principle holds then so does the following statement: Let
$x_{0},x_{1}\in U$, $y_{0},y_{1}\in V$ and $u_{0}\in J$ be given. Let
$\\{x_{\theta}\\}_{\theta\in[0,1]}$ denote the $g$-segment (in our
coordinates, line segment) from $x_{0}$ to $x_{1}$ with respect to
$y_{0},z_{0}=g^{*}(x_{0},y_{0},u_{0})$ and set
$z_{1}=g^{*}(x_{0},y_{1},u_{0})$. Then there is $M$ depending only on
$\|g\|_{C^{3}}$ such that

(61) $\displaystyle g(x_{\theta},y_{1},z_{1})-g(x_{\theta},y_{0},z_{0})\leq
M\theta[g(x_{1},y_{1},z_{1})-g(x_{1},y_{0},z_{0})]_{+}.$

We outline a short proof based on recent work of Loeper and Trudinger [23] in
the Appendix, Lemma 11 (see also [11]).

###### Proof. (Lemma 8).

We assume $K$ has been fixed small, to be chosen in the proof, and show there
is $C$ such that

(62) $Yu\left(\frac{1}{K}D\right)\subset Y\vee_{D,Ch}(0).$

To this end, fix $x\in\frac{1}{K}D$ and $y\in Yu(x)$. Let $g_{0}$ be the
corresponding support

$g_{0}(\cdot):=g(\cdot,y,g^{*}(x,y,u(x)))=g(\cdot,y,|g_{0}(0)|),$

where the second equality is because the transformed generating function
satisfies $g^{*}(0,y,u)=-u$. To prove (62) it suffices to show there is $C$
(independent of $x,y$) such that

(63) $|g_{0}(0)|\leq Ch.$

For in this case the function $g(\cdot,y,Ch)$ passes through the vertex of
$\vee=\vee_{D,Ch}$ and lies below $g_{0}$ so is nonpositive on $D$. Thus $y\in
Y\vee(0)$.

So let’s prove (63). By a Taylor series for $h(t)=g(x,y,t|g_{0}(0)|)$ there is
$C^{+},C^{-}$ (depending only on $C_{g}$) such that for any $x^{\prime}\in D$

(64) $\displaystyle g(x^{\prime},y,0)$ $\displaystyle\geq
g_{0}(x^{\prime})+C^{-}|g_{0}(0)|$ (65) $\displaystyle g(x^{\prime},y,0)$
$\displaystyle\leq g_{0}(x^{\prime})+C^{+}|g_{0}(0)|\leq C^{+}|g_{0}(0)|.$

Now let $\\{x_{\theta}\\}_{\theta\in[0,1]}$ denote the $g$-segment with
respect to $0,0$ that starts at $0$, passes through $x$, and hits $\partial D$
at some $x_{1}$. Because $x\in\frac{1}{K}D$ there is $\theta^{\prime}\leq 1/K$
with $x_{\theta^{\prime}}=x$. So (61) implies

(66) $\displaystyle g(x,y,0)\leq M\theta^{\prime}[g(x_{1},y,0)]_{+},$

where we’ve used (61) with $y_{1}=y$ and $y_{0},z_{0},u_{0}=0$ so
$g(\cdot,y_{0},z_{0})=0$. If $g(x_{1},y,0)\leq 0$, then $g(x,y,0)\leq 0$ and
we obtain (63) from (64) with $x^{\prime}=x$ (because $g_{0}$ is a support at
$x$ we have $g_{0}(x)=u(x)\geq-h$).

Otherwise, combine (66) with (64) on the left hand side and (65) on the right
hand side to obtain

$g_{0}(x)+C^{-}|g_{0}(0)|\leq M\theta^{\prime}C^{+}|g_{0}(0)|.$

Recalling $g_{0}(x)\geq-h$ and choosing $K$ large to ensure
$M\theta^{\prime}C^{+}\leq C^{-}/2$ completes the proof. ∎

##  6\. Strict convexity assuming a $g$-convex containing domain

In this section we prove the strict $g$-convexity, that is Theorem 1, by
adapting Chen and Wang’s work from the optimal transport case.

###### Proof (Theorem 1)..

We extend $u$ to $\tilde{u}$ defined on $\overline{U}$ as

$\tilde{u}(x):=\sup\\{g(x,y_{0},z_{0});g(\cdot,y_{0},z_{0})\text{ is a
$g$-support of $u$ in }\Omega\\}.$

This extension is equal to $u$ on $\Omega$ and satisfies

(67) $\displaystyle\lambda\chi_{\overline{\Omega}}\leq\det DY\tilde{u}$
$\displaystyle\leq\Lambda\chi_{\overline{\Omega}},$ (68) $\displaystyle
Y\tilde{u}(\overline{U})$ $\displaystyle=\overline{\Omega^{*}}.$

We assume work with $\tilde{u}$, though keep the notation $u$.

For a contradiction we suppose there is a support $g(\cdot,y_{0},z_{0})$ such
that

$G:=\\{x\in\overline{U};u(x)=g(x,y_{0},z_{0})\\},$

contains more than one point in $\Omega$. The first step of the proof is to
show that, after the coordinate transform with respect to $y_{0},z_{0}$, any
extreme point of the (now convex) set $G$ is in $\partial U$. The second step
is to choose a particular extreme point and obtain a contradiction from the
fact that it is in $\partial U$.

Step 1. Extreme points cannot be in the interior
Without loss of generality $y_{0},z_{0}=0$. Applying the transformation (15)
of the $x$-coordinates we have that $G$ and $U$ are convex. Assume, for a
contradiction, there is an extreme point of $G$, without loss of generality 0,
which is an interior point of $U$. After transforming the $y$ coordinates and
generating function as in Section 3 we have

$G=\\{x;u(x)=0=g(x,0,0)\\},$

and $u\geq 0$. Choose a plane $P$ that supports $G$ at $0$ and rotate so that

(69) $\displaystyle P=\\{x_{1}=0\\},\quad\quad G\cap P=\\{0\\},\quad\quad
G\subset\\{x;x_{1}\leq 0\\}\quad\text{ and }\quad-ae_{1}\in G,$

for some $a>0$. Set

$G^{h}:=\\{u<g(\cdot,0,-h)\\}.$

Because $\Omega$ is open there is $x\in G\cap\Omega$ with
$B:=B_{r}(x)\subset\Omega$ for sufficiently small $r>0$. In particular $\det
DYu\geq\lambda$ on $B$. Choose $r,h$ small enough to ensure Theorem 5 holds on
$B\cap G^{h}$ (recalling Remark 2). Then

(70) $C|G^{h}\cap B|^{2/n}\leq h.$

The $g$-convexity of $G^{h}$ (with respect to $0,-h$) implies

(71) $C|G^{h}|\leq|G^{h}\cap B|.$

where $C$ depends on $r$, $g$, and some upper bound $h_{0}\geq h$ 444For
details see [5, pg. 101] for the convex case and [27] for the $g$-convex
case.. By (70) and (71)

(72) $C|G^{h}|^{2/n}\leq h.$

To obtain a contradiction by Theorem 4 we consider section that behave like
$\\{u<t(x_{1}+a)\\}$ in the convex case. By (24) for $t$, sufficiently small,

(73) $g(-ae_{1},te_{1},0)=-at+O(t^{2})<0.$

This implies $g(\cdot,te_{1},0)<g(\cdot,te_{1},g^{*}(-ae_{1},te_{1},0))$,
since the second function is $0$ at $-ae_{1}$, i.e greater than the first.
Subsequently

$D_{t}:=\\{u<g(\cdot,te_{1},g^{*}(-ae_{1},te_{1},0))\\},$

satisfies $0\in D_{t}$, and $-ae_{1}\in\partial D_{t}$. On the other hand, the
convergence $g(\cdot,te_{1},g^{*}(-ae_{1},te_{1},0))\rightarrow 0$ as
$t\rightarrow 0$ implies

(74) $a^{+}_{t}:=\sup\\{x_{1}=x\cdot e_{1};x\in D_{t}\\}\rightarrow 0.$

Finally by the expansions in Lemma 3 and (73)

(75) $|u(0)-g(0,te_{1},g^{*}(-ae_{1},te_{1},0))|\geq Ct.$

To apply Theorem 4 we need a convex section. Transform the $x$ coordinates
according to (15) with respect to $y_{t}=te_{1},z_{t}=g^{*}(-ae_{1},y_{t},0)$.
Take the line segment joining the images under this transformation of
$-ae_{1}$ and $0$. Its length is greater than $Ca$ for a constant depending on
$g$. One of the supporting planes orthogonal to this line segment converges to
the plane $P=\\{x_{1}=0\\}$ as $t\rightarrow 0$. The other remains a distance
of at least $Ca$ from the image of $0$. Thus Theorem 4 implies, for some
$\varepsilon_{t}\rightarrow 0$, that

(76) $t\leq C\varepsilon_{t}|D_{t}|^{2/n},$

where we’ve used (75). Finally, again by (24),(27) and (73) we obtain
$D_{t}\subset G^{Ct}$ for some $C>0$ and $t$ sufficiently small. Thus (76)
with $t=h/C$ contradicts (72) and we conclude any extreme point of $G$ lies on
$\partial U$.

Step 2. An extreme point that cannot be on the boundary
Step 2a. Coordinate transform The argument in this step requires transforming
the coordinates and generating function with respect to different points so we
are explicit with the details. We begin with the original coordinates,
generating function, and contact set $G=\\{u\equiv g(\cdot,y_{0},z_{0})\\}$
which we assume contains more than one point in $\Omega$. Without loss of
generality $y_{0},z_{0}=0$. Introduce the coordinates

(77) $\tilde{x}=\frac{-g_{y}}{g_{z}}(x,0,0).$

Pick any $x_{1}\in G$ and denote its image under (77) by $\tilde{x_{1}}$.
Define

$\overline{y}=E^{-1}(x_{1},0,0)[g_{x}(x_{1},y,g^{*}(x_{1},y,u(x_{1})))-g_{x}(x_{1},0,0)].$

The $E^{-1}$ factor implies, via a Taylor series in $y$, that

(78) $\overline{y}=y+O(|y|^{2}).$

The images of $\Omega,\Omega^{*}$ in these coordinates are denoted by555In
step 2a the overline notation is used for coordinates not closures.
$\tilde{\Omega},\overline{\Omega^{*}}$. Both are convex and
$0\in\overline{\Omega^{*}}$. By a rotation, which is applied to $\tilde{x}$
and $\overline{y}$, we assume $t_{0}e_{1}\in\overline{\Omega^{*}}$ for some
small $t_{0}$. By convexity $\overline{\Omega^{*}}$ contains a cone

(79) $\mathcal{C}:=\\{t\xi;0<t\leq t_{0},\xi\in B_{r}^{\prime}\\},$

where $B_{r}^{\prime}$ is a geodesic ball in the hemisphere
$\mathbf{S}^{n-1}\cap\\{x_{1}>0\\}$ centered on $(1,0,\dots,0)$ (Figure 1).

Now choose a specific extreme point of $\tilde{G}$ as follows: Take the
paraboloid

$\tilde{P}_{M}=\\{\tilde{x}=(\tilde{x}_{1},\dots,\tilde{x}_{n});\tilde{x}_{1}=-\varepsilon(\tilde{x}_{2}^{2}+\dots+\tilde{x}_{n}^{2})+M\\}$

for large $M$ and small $\varepsilon$. Decrease $M$ until the paraboloid first
touches $\tilde{G}$, necessarily at an extreme point $\tilde{x_{0}}$. We take
$\tilde{P}$ as the tangent plane to $\tilde{P}_{M}$ at $\tilde{x_{0}}$ and
note $\tilde{P}$ supports $\tilde{G}$ at $\tilde{x_{0}}$. Provided
$\varepsilon$ is sufficiently small (depending on $\text{diam}(G)$) the normal
to $\tilde{P}$ is in $B_{r}^{\prime}$ (Figure 1).

Our final coordinate transform is

(80) $\overline{x}\mapsto-g_{z}(x_{0},y_{0},z_{0})[\tilde{x}-\tilde{x_{0}}].$

which is a dilation and translation (but no rotation) of the $\tilde{x}$
coordinates. In these coordinates the image of $U$, denoted $\overline{U}$, is
convex, and provided $\varepsilon$ was chosen small depending also on $C_{g}$
(but importantly independent of $x_{0}$), we have the normal to
$\overline{P}$, the image of $\tilde{P}$ under (80), at $0$ is still in
$B_{r}^{\prime}$. Set

$\displaystyle\tilde{g}(x,y,z)$
$\displaystyle=\frac{g_{z}(x_{0},0,0)}{g_{z}(x,0,0)}[g(x,y,g^{*}(x_{0},y,u(x_{0})-z))-g(x,0,0)],$

and define $\overline{g}(\overline{x},\overline{y},z)=\tilde{g}(x,y,z)$ for
$\overline{x},\overline{y}$ the image of $x,y$. By a Taylor series in $y,z$,
using $\tilde{g}_{y}(x,0,0)=\overline{x},\tilde{g}_{z}(x,0,0)=-1$ and (78), we
have

(81)
$\overline{g}(\overline{x},\overline{y},z)=\overline{x}\cdot\overline{y}-z+O(|\overline{y}|^{2})+O(|\overline{y}||z|)+O(|z|^{2})$

Finally we rotate the $\overline{x}$ and $\overline{y}$ coordinates so that
the supporting plane $\overline{P}$ becomes

$\overline{P}=\\{\overline{x}=(\overline{x}_{1},\dots,\overline{x}_{n});\overline{x}_{1}=0\\},$

and $\overline{G}\subset\\{\overline{x};\overline{x}_{1}\leq 0\\}$. Because
the normal to $\overline{P}$ was in $B_{r}^{\prime}$ after this rotation we
still have $\overline{y_{0}}^{\prime}:=t_{0}e_{1}\in\overline{\Omega^{*}}$.
This is by inclusion of the cone (79) (Figure 1). Finally pick any
$\overline{x_{0}^{\prime}}\in\overline{G}$ such that
$\overline{x_{0}^{\prime}}\cdot\overline{y_{0}^{\prime}}\neq 0$. We are in
exactly the setting to use the transformation (4.7) from [4] which preserves
that $\overline{g}$ has the form (81) and after which we have

$\displaystyle\overline{G}\subset\\{\overline{x};\overline{x}_{1}\leq 0\\},$
$\displaystyle\overline{G}\cap\\{\overline{x}_{1}=0\\}=0,$ $\displaystyle-
ae_{1}\in\overline{G},$ $\displaystyle be_{1}\in\overline{\Omega^{*}}.$

Step 2b takes place in this setting. For ease of notation we drop the
overline.

$x_{1}$$\overline{\Omega^{*}}$$t_{0}e_{1}$$\mathcal{C}$$0$$\tilde{G}$$\tilde{P}_{M}:\tilde{x_{1}}=-\varepsilon(x_{1}^{2}+\dots
x_{n}^{2})+M$$\tilde{x_{0}}$$\tilde{P}$ Figure 1. The choice of $\overline{y}$
coordinates implies $\overline{\Omega^{*}}$ contains the cone $\mathcal{C}$.
By choosing $\varepsilon$ sufficiently small a normal to the paraboloid at the
extreme point lies in the cone. Thus after a rotation so $\tilde{P}$ becomes
$\overline{P}=\\{\overline{x};\overline{x}_{1}=0\\}$, $\overline{\Omega^{*}}$
still contains $t_{0}e_{1}$.

Step 2b. Obtaining the contradiction in this setting
We see $F_{\tau}:=G\cap\\{x_{1}\geq-\tau\\}$ decreases to $\\{0\\}$ as
$\tau\rightarrow 0$. Because $\Omega$ lies a positive distance from
$0\in\partial U$, for $\tau$ fixed sufficiently small
$\text{dist}(\Omega,F_{\tau})>0$. On the other hand, provided in addition,
$\tau<-a/2$ the set

(82) $\\{u<g(\cdot,te_{1},g^{*}(-\tau e_{1},te_{1},u(-\tau e_{1}))\\},$

decreases to $F_{\tau}$ as $t\rightarrow 0$. This is because by calculations
similar to part 1

$g(x,te_{1},g^{*}(-\tau e_{1},te_{1},u(-\tau e_{1}))=(x_{1}+\tau)t+O(t^{2}).$

Thus for $t$ small the set in (82) is disjoint from $\overline{\Omega}$. So
for all $x\in\overline{\Omega}$,

(83) $u(x)>g(x,te_{1},g^{*}(-\tau e_{1},te_{1},u(-\tau e_{1})).$

On the other hand $te_{1}\in\Omega^{*}$ for $t$ small. Thus there is
$x_{2}\in\overline{\Omega}$ with $te_{1}\in Yu(x_{2})$. So

(84) $u(x)\geq g(x,te_{1},g^{*}(x_{2},te_{1},u(x_{2}))),$

for all $x\in U$. For a contradiction evaluate (84) at $x=-\tau e_{1}$, then
apply

$g(x_{2},te_{1},g^{*}(-\tau e_{1},te_{1},\cdot))$

to both sides. The resulting inequality contradicts (83) with $x=x_{2}$ and
thereby completes the proof. ∎

######  Remark 3.

A corollary of Theorem 1 is that when $\Omega$ is uniformly $g$-convex with
respect to $u$ and $\overline{\Omega}\subset U$ then $u$ is strictly
$g$-convex: no convexity condition is needed on $U$. This is immediate; for
$\varepsilon$ sufficiently small
$\Omega_{\varepsilon}:=\\{x;\text{dist}(x,\Omega)<\varepsilon\\}$ is uniformly
$g$-convex with respect to $u$ and strictly contains $\Omega$.

It would be desirable to prove the strict convexity under the extension of the
hypotheses of Figalli, Kim, McCann [6]. These hypotheses are strict
$c$-convexity of both domains, which is stronger than here — however they only
require the cost function be defined on
$\overline{\Omega}\times\overline{\Omega^{*}}$. Unfortunately when one
attempts to extend their crucial Theorem 5.1 a $g_{i,j,z}$ term appears in the
analogue of their equation (5.2) and prohibits a similar proof.

##  7\. $C^{1}$ differentiability of strictly $g$-convex solutions

Once the strict convexity has been proved we obtain the $C^{1}$
differentiability using similar techniques.

######  Theorem 6.

Assume $g$ satisfies LMP and $\lambda,\Lambda>0$. Suppose
$u:\Omega\rightarrow\mathbf{R}$ is a strictly $g$-convex Aleksandrov solution
of $\lambda\leq\det DYu\leq\Lambda.$ Then $u\in C^{1}(\Omega)$.

###### Proof.

Suppose for a contradiction at some $x_{0}$, assumed to be $0$, $\partial
u(0)$ contains more than one point. Let $p_{0}$ be an extreme point of
$\partial u(0)$ with $y_{0}:=Y(0,u(0),p_{0})$ and $g(\cdot,y_{0},z_{0})$ the
corresponding support. Without loss of generality $u(0),y_{0}=0$. Fix $h>0$
small, and apply the transformations in Section 3 so $g$ satisfies (23) and
$u(0)=-h$. We set $G^{h}=\\{u<0=g(\cdot,0,0)\\},$ which is a section and thus
convex.

After these transformations, by (23) and (27), $p=g_{x}(0,0,h)$ is an extreme
point of $\partial u(0)$ satisfying $|p|=O(h^{2})$. Thus after subtracting
$-h+p\cdot x$ from both $u$ and the generating function we have

(85) $\\{u<h/2\\}\subset G^{h}=\\{u<h-p\cdot x\\}\subset\\{u<3h/2\\}.$

We’ve used that by the strict convexity and an initial choice of the section
small $|x|$ is as small as desired. Moreover, $0$ is now an extreme point of
$\partial u(0)$ and we assume, after a rotation, for some small $a>0$

$\displaystyle\\{p;p_{1}\geq 0\\}$ $\displaystyle\supset\partial u(0)$ (86)
$\displaystyle ae_{1}$ $\displaystyle\in\partial u(0),$

for some small $a>0$. This implies $u(-te_{1})=o(t)$. Thus $\\{u<h/2\\},$
contains $-R(h)he_{1}$ for some positive function $R$ satisfying
$R(h)\rightarrow\infty$ as $h\rightarrow 0$ (for details, see [27, Lemma
A.4]).

We recall inequality 41, so that for $x$ in a small neighbourhood of $0$ we
have

$u\geq g(x,Y(0,ae_{1},u(0)),h)\geq\frac{x_{1}a}{2}-Ch,$

where we’ve used $Y(0,ae_{1},u(0))=ae_{1}+O(h)$. So for $h$ small, the second
subset relation in (85) implies $\sup_{G^{h}}x_{1}\leq Ch/a$.

Thus Theorem 4 implies

$h\leq\frac{C}{R(h)}|G^{h}|^{2/n}.$

On the other hand Theorem 5 yields $|G^{h}|^{2/n}\leq Ch$, which is a
contradiction as $h\rightarrow 0$. ∎

##  8\. Global regularity

In this section we use the strict $g$-convexity to prove the global regularity
of Aleksandrov solutions of the second boundary value problem.

To prove Theorem 2 it suffices to prove there is $v\in
C^{3}(\overline{\Omega})$ solving (13) subject to (4) and satisfying
$v(x_{0})=u(x_{0})$ for some $x_{0}\in\Omega.$ Here’s why. Our strict
$g$-convexity result, Remark 3, implies the Aleksandrov solution is strictly
$g$-convex. Then by recent work of Trudinger [30, Theorem 3.4], $u\in
C^{3}(\Omega)$. Thus $u,v$ are regular enough to apply the uniqueness result,
[26, Theorem 1.1], and conclude $u=v$ so $u\in C^{3}(\overline{\Omega})$.

The existence of the desired $v$ is proved by modifying Jiang and Trudinger’s
[14] global existence result so as to construct a solution taking a prescribed
value at a given point. For the Monge–Ampère and optimal transport cases this
is trivial — just add or subtract a constant. In our case we modify Jiang and
Trudinger’s use of degree theory to obtain a solution that is close to the
prescribed value at a given point. Taking a limit gives the desired globally
smooth solution. We note apart from a slightly more restrictive height
condition (which ensures the constructed function doesn’t leave $J$), our
hypotheses are those from the existence theory [14, Theorem 1.1].

For the degree theory we use that under the hypotheses of Theorem 2 the second
boundary value problem can be rewritten as a Monge–Ampère type equation [29]
coupled with a uniformly oblique boundary condition [14]. That is
$C^{4}(\Omega)\cap C^{3}(\overline{\Omega})$ solutions of (13) subject to (4)
solve

(87) $\displaystyle\det[D^{2}u-A(\cdot,u,Du)]$
$\displaystyle=B(\cdot,u,Du),\text{ in }\Omega$ (88) $\displaystyle
G(\cdot,u,Du)$ $\displaystyle=0\text{ on }\partial\Omega$

where

$\displaystyle A(\cdot,u,Du)$
$\displaystyle=g_{ij}(\cdot,Y(\cdot,u,Du),Z(\cdot,u,Du))$ $\displaystyle
B(\cdot,u,Du)$ $\displaystyle=\det E\frac{f(\cdot)}{f^{*}(Y(\cdot,u,Du))},$

and $G(x,u,p)$ satisfies

$G_{p}(x,u,Du)\cdot\gamma\geq c_{0}>0\text{ on }\partial\Omega,$

for $\gamma$ the outer unit normal to $\partial\Omega$ and $c_{0}$ independent
of $u$ [14]. Written in this way we introduce the following additional
assumptions used by Jiang and Trudinger for the existence theory, and used
here for the regularity theory.

A4w.  The matrix $A$ satisfies

$D_{u}A_{ij}(x,u,p)\xi_{i}\xi_{j}\geq 0,$

for all $(x,u,p)\in\mathcal{U}$ and $\xi\in\mathbf{R}^{n}$.

A5.  Assume $u:\Omega\rightarrow\mathbf{R}$ is a $g$-convex function. There is
$K_{0}$ depending on $g$ and $Yu(\Omega)$ such that for all $x\in\Omega,y\in
Yu(x)$, and $z=g^{*}(x,y,u(x))$, there holds $|g_{x}(x,y,z)|\leq K_{0}$.

The degree theory serves as a high-powered version of the method of
continuity. As in the method of continuity we need apriori estimates and a
smooth function solving a problem in the same homotopy class. These are
provided by the following results from the literature.

######  Theorem 7.

[13, Theorem 3.1] Assume $g$ is a $C^{4}$ generating function satisfying LMP.
Assume $u\in C^{4}(\Omega)\cap C^{3}(\overline{\Omega})$ is a $g$-convex
solution of (13) subject to (4). Then provided $f\in
C^{2}(\overline{\Omega}),f^{*}\in C^{2}(\overline{\Omega^{*}})$ satisfy the
mass balance condition (14) and $\Omega,\Omega^{*}$ are respectively uniformly
$g/g^{*}$-convex with respect to $u$, there is $C>0$, depending on
$A,B,\Omega,\Omega^{*}$ and $\|u\|_{C^{1}(\Omega)}$ such that

$\|u\|_{C^{2}}(\Omega)\leq C.$

######  Remark 4.

If $\|u\|_{C^{1}(\Omega)}$ is controlled independently of $u$, then the
$C^{2}$ estimate is independent of $u$. Subsequently when $u$ has higher order
derivatives, estimates for these follow by the elliptic theory. More precisely
$C^{2,\alpha}(\overline{\Omega})$ estimates from [18, Theorem 1] then
$C^{4,\alpha}(\overline{\Omega})$ estimates by the linear theory. The use of
the linear theory is standard and outlined in [27, Lemma 7.7]. For the
$C^{4,\alpha}$ estimates, as in [14], we smooth our domains so they are
$C^{5}$. However uniform $C^{3,\alpha}$ estimates are independent of this
smoothing.

######  Lemma 9.

[14, Lemma 2.3] Fix $x_{0}\in\Omega,y_{0}\in\Omega^{*}$, $z_{0}\in
I_{x_{0},y_{0}}$ and $g$-convex $g_{0}(\cdot):=g(\cdot,y_{0},z_{0})$. Assume
$\Omega,\Omega^{*}$ are uniformly $g/g^{*}$-convex with respect to $g_{0}$.
Define

$\Omega_{\delta}=\\{x\in U;\text{dist}(x,\Omega)<\delta\\}.$

Then for all $0<\varepsilon<<\delta$ sufficiently small there is a smooth
uniformly $g$-convex $u_{\varepsilon}\in
C^{\infty}(\overline{\Omega_{\delta}})$ and a smooth uniformly $g^{*}$-convex
domain $\Omega^{*}_{\delta,\varepsilon}$ satisfying the following convergence
properties

$\displaystyle\Omega^{*}_{\delta,\varepsilon}\rightarrow\Omega^{*}_{\delta,0}\text{
in the Hausdorff distance as }\varepsilon\rightarrow 0$
$\displaystyle\Omega^{*}_{\delta,0}\rightarrow\Omega^{*}\text{ in the
Hausdorff distance as }\delta\rightarrow 0$

and

(89) $g(\cdot,y_{0},z_{0})\leq u_{\varepsilon}\leq
g(\cdot,y_{0},z_{0})+C\varepsilon,$

with

$Yu_{\varepsilon}(\Omega_{\delta})=\Omega^{*}_{\delta,\varepsilon}.$

The uniform $g^{*}$-convexity of is $\Omega^{*}_{\delta,\varepsilon}$ is with
respect to $u_{\varepsilon},$ and $\Omega_{\delta}$ is uniformly $g$-convex
with respect to $u_{\varepsilon}$.

The following lemma, in particular condition (92) is what allows us to prove
the global regularity. We let $x_{0},g(\cdot,y_{0},z_{0})$ be as in Theorem 2
and use the notation from Lemma 9.

######  Lemma 10.

Assume the hypothesis of Theorem 2. There exists $v\in
C^{3,\alpha}(\overline{\Omega}_{\delta})$ depending on $\delta,\varepsilon$
and satisfying

(90) $\displaystyle\det DY(\cdot,v,Dv)$
$\displaystyle=\frac{f_{\delta}(\cdot)}{f_{\varepsilon}^{*}(Y(\cdot,v,Dv))}\text{
in }\Omega_{\delta}$ (91) $\displaystyle Yv(\Omega_{\delta})$
$\displaystyle=\Omega^{*}_{\delta,\varepsilon}$ (92) $\displaystyle v(x_{0})$
$\displaystyle=u_{\varepsilon}(x_{0})$

###### Proof.

Within this proof we denote $f,f^{*},\Omega,\Omega^{*}$, and $u_{0}$ by
$f_{\delta},f^{*}_{\varepsilon},$
$\Omega_{\delta},\Omega^{*}_{\delta,\varepsilon},$ and $u_{\varepsilon}$. Fix
a smooth cutoff function $\eta$ for the unit ball, i.e. $\eta>0$ on $B_{1}(0)$
and $\eta=0$ on $\overline{B_{1}}^{c}$. For $a<<1$ set
$\eta_{a}(x)=a^{4}\eta((x-x_{0})/a)$ where the $a^{4}$ term ensures
$\|\eta_{a}\|_{C^{4}}\leq C(\eta)$.

For $t\in[0,1]$ and $\tau$ to be fixed large we consider the family of
problems

(93) $\displaystyle f^{*}(Y(\cdot,v,Dv))$ $\displaystyle\det
DY(\cdot,v,Dv)=e^{[(1-t)\tau+\eta_{a}(\cdot)](v-u_{0})}\big{[}tf(\cdot)$
$\displaystyle\quad+(1-t)f^{*}(Y(\cdot,u_{0},Du_{0}))\det
DY(\cdot,u_{0},Du_{0})\big{]}$ (94) $\displaystyle Yv(\Omega)$
$\displaystyle=\Omega^{*}.$

We restrict our attention to $g$-convex solutions $v\in
C^{4,\alpha}(\overline{\Omega})$. For $t=0$ the problem is solved by $u_{0}$.
We aim to show, using the degree theory of Li, Liu, and Nguyen [17], that this
problem has a solution for $t=1$. The required
$C^{4,\alpha}(\overline{\Omega})$ bounds hold by Remark 4 provided we obtain a
$C^{1}$ estimate. Then the degree theory is applied as follows. By [17,
Theorem 1 (p2)] the problems (93) and (94) have the same degree for $t=0$ and
$t=1$. Then by [17, Corollary 2.1 (a)] to show a solution exists for $t=1$ it
suffices to show the problem at $t=0$ has non-zero degree. Finally by a
combination of Corollary 2.1 (d) and Theorem 1 (p3) from the same paper, the
problem has non-zero degree at $t=0$ provided both the problem for $t=0$ has a
unique solution and the linearized problem at $t=0$ (linearized about $u_{0}$)
is uniquely solvable.

Step 1. a-priori $\|v\|_{C^{1}}$ estimates We show using the mass balance
condition that any solution of (93) subject to (94) intersects $u_{0}$ in
$\Omega$. Furthermore, for $t=1$ this intersection occurs in $B_{a}(x_{0})$.
This will allow us to send $a\rightarrow 0$ and obtain (92). Assume to the
contrary $v>u_{0}$ on $\Omega$. The proof is similar if $v<u_{0}$ on $\Omega$.
Because $v>u_{0}$ we have $e^{[(1-t)\tau+\eta_{a}(\cdot)](v-u_{0})}>1$. Now
(93) and (94) along with mass balance and the change of variables formula
yield the following contradiction

$\displaystyle\int_{\Omega^{*}}f^{*}$
$\displaystyle=\int_{\Omega}f^{*}(Y(\cdot,v,Dv))\det DY(\cdot,v,Dv)$
$\displaystyle>\int_{\Omega}tf(\cdot)+(1-t)f^{*}(Y(\cdot,u_{0},Du_{0}))\det
DY(\cdot,u_{0},Du_{0})$ $\displaystyle=\int_{\Omega^{*}}f^{*}.$

For $t=1$ we obtain the same contradiction if $v>u_{0}$ on $B_{a}$ for in this
case

$\displaystyle f^{*}(Y(\cdot,v,Dv))\det DY(\cdot,v,Dv)$ $\displaystyle>f\text{
on }B_{a}(x_{0})$ $\displaystyle f^{*}(Y(\cdot,v,Dv))\det DY(\cdot,v,Dv)$
$\displaystyle=f\text{ on }\Omega\setminus\overline{B_{a}(x_{0})}.$

An estimate $\sup_{\Omega}|Dv|\leq C$ follows by A5 provided
$v(\overline{\Omega})\subset J$. This inclusion follows by using contact point
$v(x^{\prime})=u_{0}(x^{\prime})$, the inequality (89), and the condition on
$g(\cdot,y_{0},z_{0})$ in the statement of the Theorem 2. Therefore
$\|v\|_{C^{1}(\overline{\Omega})}\leq C$.

Step 2. Unique solvability of the linear and nonlinear problem for $t=0.$
First note the nonlinear problem has a unique solution by [26, Theorem 1.1].
The linearisation at $t=0$ is a problem of the form

(95) $\displaystyle a^{ij}D_{ij}v+b^{i}D_{i}v-cv$ $\displaystyle=0\text{ in
}\Omega$ $\displaystyle\alpha v+\beta\cdot Dv$ $\displaystyle=0\text{ on
}\partial\Omega,$

where $a^{ij},b^{i},\alpha,\beta$ depend on $u_{0}$ and $g$, the equation is
uniformly elliptic, $\beta$ is a strictly oblique vector field (by [14], see
also [27, Theorem 7.4]) and, provided $\tau$ is sufficiently large,
$c>\tau/2$. We show this problem has the unique solution $v=0$ (thus, by the
Fredholm Alternative, also proving the existence of a solution). If not,
assume $v$ is a solution positive at some point in $\overline{\Omega}$. Take a
defining function $\varphi$ for $\Omega$, satisfying $D\varphi=\gamma$ (the
outer unit normal), and set $w=e^{-\kappa\varphi}v$ for large $\kappa$ to be
chosen. If the positive maximum of $w$ occurs at $x_{1}\in\partial\Omega$ then
by the obliqueness

$0\leq\beta\cdot Dw(x_{1})=e^{-\kappa\varphi}[\beta\cdot
Dv-\kappa(\beta\cdot\gamma)v].$

With the linearised boundary condition we obtain

$\kappa(\beta\cdot\gamma)v(x_{1})\leq\beta\cdot Dv(x_{1})=-\alpha v(x_{1}),$

a contradiction for $\kappa$ sufficiently large depending only on
$\alpha$,$c_{0}$. Thus $w$ attains its positive maximum at $x_{1}\in\Omega$.
At this point

$\displaystyle 0$ $\displaystyle\geq e^{\kappa\varphi}a^{ij}D_{ij}w(x_{1})$
(96) $\displaystyle=a^{ij}D_{ij}v-2\kappa a^{ij}D_{i}\varphi D_{j}v-\kappa
va^{ij}D_{ij}\varphi+\kappa^{2}va^{ij}D_{i}\varphi D_{j}\varphi.$

At an interior maximum $Dw(x_{1})=0$ so $D_{i}v(x_{1})=\kappa
D_{i}\varphi(x_{1})v(x_{1})$. Thus we can eliminate $Dv$ terms in (96). In
combination with (95) and $c>\tau/2$ we obtain

$0\geq e^{\kappa\varphi}a^{ij}D_{ij}w(x_{1})\geq C(\tau-C_{1})v(x_{1}),$

for $C_{1}$ independent of $\tau$. This is a contradiction for $\tau$
sufficiently large.

Conclusion. By the degree theory there is $v\in
C^{4,\alpha}(\overline{\Omega})$ solving (93) subject to (94) for $t=1$. Our
$C^{3,\alpha}$ estimates are independent of $a$ and the domain smoothing. Thus
we consider a sequence of $a_{k}\rightarrow 0$ and the corresponding solutions
at $t=1$ denoted $v_{a_{k}}\in C^{3,\alpha}(\overline{\Omega})$. By
Arzela–Ascoli we have uniform convergence to some $v\in C^{3,\alpha}(\Omega)$
solving (93) for $t=1$ and (94). Noting there is $x_{k}\in B_{a_{k}}(x_{0})$
with $v_{a_{k}}(x_{k})=u_{0}(x_{k})$ we conclude by the uniform convergence
that the limiting function $v$ satisfies $v(x_{0})=u_{0}(x_{0})$. ∎

To complete the proof of Theorem 2 take the solution of (90)-(92) and note the
$C^{3,\alpha}$ bounds are independent of the parameters $\varepsilon,\delta$
(provided these parameters are initially fixed small). Send first
$\varepsilon$ then $\delta$ to $0$ and obtain a $C^{3}(\overline{\Omega})$
solution of (1) subject to (4) satisfying $u(x_{0})=v(x_{0})$, thereby
completing the proof of Theorem 2 outlined at the start of this section.

##  Appendix A Omitted proofs

###### Proof. Theorem 2.

Point (2) is immediate by direct calculation (verify directly (7) and (8)).
The first point is also by direct calculation, but more involved. To begin,
note

(97) $\displaystyle\tilde{g}_{x}(x,y,z)$
$\displaystyle=\frac{-g_{xz}(x,0,0)}{g_{z}(x,0,0)}\tilde{g}(x,y,z)$
$\displaystyle\quad\quad+\frac{g_{z}(0,0,0)}{g_{z}(x,0,0)}[g_{x}(x,y,g^{*}(0,y,h-z))-g_{x}(x,0,0)],$

and

(98)
$\displaystyle-\frac{\tilde{g}_{y}}{\tilde{g}_{z}}(x,y,z)=\frac{1}{g^{*}_{u}(0,y,h-z)}\frac{g_{y}}{g_{z}}(x,y,g^{*}(0,y,h-z))+\frac{g^{*}_{y}}{g^{*}_{u}}(0,y,h-z).$

Thus the A1 condition for $\overline{g}$ holds because, by (97), for fixed $x$
the mapping $(y,z)\mapsto(\tilde{g}(x,y,z),\tilde{g}_{x}(x,y,z))$ is injective
(by the A1 condition for $g$). Similar reasoning using (98) yields the A1∗
condition for $\overline{g}$. More precisely by (98) for fixed $(y,z)$ the
mapping $x\mapsto\frac{\tilde{g}_{y}}{\tilde{g}_{z}}(x,y,z)$ is injective by
A1∗.

Next, we introduce the notation $\overline{Y}(q,U,P),\overline{Z}(q,U,P)$ to
denote $\overline{Y},\overline{Z}$ solving

$\displaystyle\overline{g}(q,\overline{Y}(q,U,P),\overline{Z}(q,U,P))=U,$
$\displaystyle\overline{g}_{q}(q,\overline{Y}(q,U,P),\overline{Z}(q,U,P))=P.$

We compute

$\displaystyle\overline{Z}(q,U,P)=h-$ $\displaystyle
g\left[0,\overline{Y}(x,U,P),g^{*}\left(x,\overline{Y}(x,U,P),\frac{g_{z}(x,0,0)}{g_{z}(0,0,0)}U+g(x,0,0)\right)\right]$
(99) $\displaystyle\overline{Y}(q,U,P)$
$\displaystyle=p\Big{[}Y\Big{(}x,\frac{g_{z}(x,0,0)}{g_{z}(0,0,0)}U+g(x,0,0),$
$\displaystyle\quad\quad\frac{g_{z}(x,0,0)}{g_{z}(0,0,0)}\frac{\partial
q}{\partial x}P+\frac{g_{x,z}(x,0,0)}{g_{z}(0,0,0)}U+g(x,0,0)\Big{)}\Big{]}.$

For the A2 condition first note the calculation $\tilde{g}_{z}<0$ implies
$\overline{g}_{z}<0$. To check $\det\overline{E}\neq 0$ it suffices to check
$\det D_{P}\overline{Y}\neq 0$ [29, eq. 2.2]. This follows from (99). The key
point is that, despite the unwieldy expression, we have
$\overline{Y}(q,U,P)=p(Y(x,l_{1}(U),l_{2}(P)))$ for a nonsingular linear
function $l_{2}(P)$.

For the Loeper maximum principle we must verify to verify for each
$q,q^{\prime},U$

$\displaystyle\overline{g}(q^{\prime},$
$\displaystyle\overline{Y}(q,U,P_{\theta}),\overline{Z}(q,U,P_{\theta}))$
$\displaystyle\leq\max\\{\overline{g}(q^{\prime},\overline{Y}(q^{\prime},U,P_{0}),\overline{Z}(q,U,P_{0})),\overline{g}(q,\overline{Y}(q,U,P_{1}),\overline{Z}(q,U,P_{1}))\\},$

whenever $\\{P_{\theta}\\}_{\theta\in[0,1]}$ is a line segment for which the
above quantities are well defined. This follows by a direct calculation using
the definition of $\overline{g}$, (99), and the Loeper maximum principle for
$g$. ∎

######  Lemma 11.

Assume $g$ is a $C^{3}$ generating function satisfying the Loeper maximum
principle. Then the statement of quantitative-quasiconvexity (61) holds.

###### Proof.

We’ll use the notation preceding (61). We note by recent work of Loeper and
Trudinger [23] the Loeper maximum principle implies the well known A3w
condition in the sense that the function

$p\mapsto A_{ij}(x,u,p)\xi_{i}\xi_{j},$

is convex on line segments orthogonal to $\xi$. Thus, working in the
transformed $x$ coordinates with respect to $y_{0},z_{0}$, we define the
$g$-segment $x_{\theta}=\theta x_{1}+(1-\theta)x_{0}$ and put

$h(\theta):=g(x_{\theta},y_{1},z_{1})-g(x_{\theta},y_{0},z_{0}).$

Guillen and Kitagawa [9, Lemma 9.3] prove when $g$ is $C^{4}$ and satisfies
A3w

(100) $h^{\prime\prime}(\theta)\geq-K|h^{\prime}(\theta)|.$

We prove (100) holds for $g\in C^{3}(\Gamma)$ satisfying A3w in the above
sense by a minor modification of the proofs in [29, 2.11], [28, Appendix]
which are for $g\in C^{4}(\Gamma)$. In the proofs op. cit. $C^{4}$
differentiability is used to deal with a term

$\big{[}A_{ij}(x_{\theta},u_{0},p_{1})-A_{ij}(x_{\theta},u_{0},p_{0})-D_{p_{k}}A_{ij}(x_{\theta},u_{0},p_{0})(p_{1}-p_{0})\big{]}(x_{1}-x_{0})_{i}(x_{1}-x_{0})_{j}.$

where $p_{1}=g_{x}(x_{\theta},y_{1},z_{1})$ and
$p_{0}=g_{x}(x_{\theta},y_{0},z_{0})$. For merely $C^{3}$ $g$ we set
$\overline{x}=x_{1}-x_{0}$, $\overline{p}=p_{1}-p_{0}$ and compute, by a
Taylor series for
$A_{ij}(x_{\theta},u_{0},p_{1}-(1-t)(\overline{p}\cdot\overline{x})\overline{x}/|\overline{x}|^{2})$,
the inequality

$\displaystyle\big{[}A_{ij}(x_{\theta},u_{0},p_{1})-A_{ij}(x_{\theta},u_{0},p_{0})-D_{p_{k}}A_{ij}(x_{\theta},u_{0},p_{0})\overline{p}\big{]}\overline{x}_{i}\overline{x}_{j}$
$\displaystyle\geq\big{[}A_{ij}(x_{\theta},u_{0},p_{1}-(\overline{p}\cdot\overline{x})\overline{x}/|\overline{x}|^{2})-A_{ij}(x_{\theta},u_{0},p_{0})$
$\displaystyle\quad\quad-
D_{p_{k}}A_{ij}(x_{\theta},u_{0},p_{0})(\overline{p}-(\overline{p}\cdot
x)\overline{x}/|\overline{x}|^{2})\big{]}\overline{x}_{i}\overline{x}_{j}-K\overline{p}\cdot\overline{x}.$

Here $K$ depends on $|D_{p}A_{ij}|$, i.e. $\|g\|_{C^{3}}$. Noting the line
segment from $p_{0}$ to
$p_{1}-(\overline{p}\cdot\overline{x})\overline{x}/|\overline{x}|^{2}$ is
orthogonal to $\overline{x}$ we obtain (100) by the A3w condition and
$\overline{p}\cdot\overline{x}=h^{\prime}(\theta)$. Then (61) is a consequence
of (100) [9, Corollary 9.4]. ∎

For cost functions this is proved (under weaker assumptions) by Jeong [10].
The brevity of our proof is superficial: the real work is hidden in Loeper and
Trudinger’s result that LMP implies A3w for $C^{2}$ generating functions.

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