On the convexity theory of generating functions

1 Introduction

Generating functions were introduced by the second author in [1] as nonlinear extensions of affine functions in Euclidean space for the purpose of extending the framework of optimal transportation to embrace near field geometric optics. Regularity and classical existence results depend on an underlying convexity theory which is of interest in its own right. Following our previous treatment in the optimal transportation case [2], we develop in this paper the convexity theory under minimal smoothness assumptions on the generating function. As in [2], the approach is largely geometric, somewhat shadowing that in [3], and as a byproduct we obtain a completely different derivation of the invariance of our conditions under duality, from the complicated analytic calculation in [1].

A generating function can be defined on the product of two Riemannian manifolds and the real line. Here as in [1] we will restrict attention to the Euclidean space case so that our generating functions are defined on domains Γ ⊂ R n × R n × R , whose projections,

are open intervals. Assuming g ∈ C ²(Γ), g _z ≠ 0 in Γ, normalised so that g _z < 0 in Γ, and denoting

we then have the following two fundamental conditions from [1],

• A1: For each ( x , u , p ) ∈ U , there exists a unique point (x, y, z) ∈ Γ satisfying

  g ( x , y , z ) = u , g x ( x , y , z ) = p .

• A2: det E ≠ 0, in Γ, where E is the n × n matrix given by

  E = [ E i , j ] = g x , y − ( g z ) − 1 g x , z ⊗ g y .

  In the special case of optimal transportation,

  (1.2) g ( x , y , z ) = − c ( x , y ) − z , Γ = D × R , g z = − 1 , I ( x , y ) = R , E = − c x , y ,

  where D is a domain in R n × R n and c ∈ C 2 ( D ) is a cost function, satisfying conditions A1 and A2 in [4].

  By defining Y(x, u, p) = y and Z(x, u, p) = z in A1, the mapping Y together with the dual function Z are generated by the equations

  (1.3) g ( x , Y , Z ) = u , g x ( x , Y , Z ) = p .

  Since the Jacobian determinant of the mapping (y, z) → (g _x , g)(x, y, z) is g _z  det E, ≠ 0 by A2, the functions Y and Z are C ¹ smooth. By differentiating (1.3) with respect to p, we also have Y _p = E ⁻¹.

  Our next fundamental condition is expressed in terms of the matrix function A on U , given by

  (1.4) A ( x , u , p ) = g x x ( x , Y ( x , u , p ) , Z ( x , u , p ) ) ,

  and extends condition G3w in [1] to non differentiable A. For its formulation we use the notation U ( x , u ) to denote the projection { p ∈ R n | ( x , u , p ) ∈ U } .

• A3w: The matrix function A is co-dimension one convex in U , with respect to p, in the sense that the function (Aξ, ξ)(x, u, ⋅) is convex along line segments in U ( x , u ) orthogonal to ξ, for all ξ ∈ R n , ( x , u ) ∈ R n × R .

  We also have a strict version of condition A3w, extending condition G3 in [1], namely

• A3s: The matrix function A is locally uniformly co-dimension one convex in U , with respect to p, in the sense that the function (Aξ, ξ)(x, u, ⋅) is uniformly convex along closed line segments in U ( x , u ) orthogonal to ξ, for all ξ ∈ R n , ( x , u ) ∈ R n × R .

  Note that here we call a function f : I → R on a closed interval I uniformly convex if the function, t → f(t) − δt ², is convex on I for some positive constant δ.

  When A is twice differentiable in p then conditions A3w, (A3s), can be expressed as

  (1.5) ( D p k p l A i j ) ξ i ξ j η k η l ≥ 0 , ( > 0 ) ,

  in U , for all ξ , η ∈ R n such that ξ ⋅ η = 0.

  Historically these conditions arose from condition A3 for local regularity in optimal transportation introduced in [4], with the weak version A3w subsequently introduced in [5], [6] for global regularity. We will express them in the non smooth case more precisely in Section 2 and moreover show that they can also be formulated for generating functions g ∈ C ¹(Γ) satisfying just condition A1, corresponding to the optimal transportation case in [7], where it is also shown that condition A3w, in the smooth case (1.5), is necessary for the regularity and convexity theories.

  The strict monotonicity property of the generating function g with respect to z, enables us to define a dual generating function g*,

  (1.6) g ( x , y , g * ( x , y , u ) ) = u ,

  with (x, y, u) ∈ Γ* ≔ {(x, y, g(x, y, z))|(x, y, z) ∈ Γ}, g x * = − g x / g z , g y * = − g y / g z and g u * = 1 / g z , which leads to a dual condition to A1, which is also critical for our convexity theory, namely

• A1*: The mapping Q: = −g _y /g _z is one-to-one in x, for all (y, z) such that (x, y, z) ∈ Γ.

  Since the Jacobian matrix of the mapping x → Q(x, y, z) is −E ^t /g _z where E ^t is the transpose of E, its determinant will not vanish when condition A2 holds, that is A2 is self dual. We will prove the invariance of conditions A3w and A3s under duality from the local convexity theory in Section 2, which also provides an alternative proof of the case g ∈ C ⁴(Γ) in [1], which is done there through explicit calculation of D _pp A. Note that by setting

  P ( x , y , u ) = g x x , y , g * ( x , y , u ) ,

  we may also express condition A1 in the same form as A1*, namely the mapping P is one-to-one in y, for all (x, u) such that (x, y, u) ∈ Γ*.

2 Local convexity

We recall the definition from [1] that a domain Ω is g-convex, (uniformly g-convex), with respect to ( y 0 , z 0 ) ∈ R n × R , if (Ω, y ₀, z ₀) ⊂ Γ and the image Q ₀(Ω) ≔ Q(⋅, y ₀, z ₀)(Ω) is convex, (uniformly convex), in R n . In this section, we will examine the relationship between local g-convexity at boundary points of g-sections and condition A3w. First we note that we can write condition A3w in the form:

for any ( x 0 , u 0 , [ p 0 , p 1 ] ) ⊂ U , p _θ = (1 − θ)p ₀ + θp ₁, 0 ≤ θ ≤ 1 and ξ·(p ₁ − p ₀) = 0. Here and throughout we use the notation [p ₀, p ₁] to denote the closed straight line segment joining points p ₀ and p ₁ in R n .

Defining now

for x ∈ Ω 0 = U ( u 0 , [ p 0 , p 1 ] ) : = { x | ( x , u 0 , [ p 0 , p 1 ] ) ∈ U } , θ ∈ (0, 1], we see that (2.1) can be written as

for all ξ ∈ R n such that ξ·Dh _θ (x ₀) = 0. This leads to the following geometric interpretation of condition A3w. Namely for the section S _θ = {x ∈ Ω₀|h _θ (x) < 0}, the second fundamental form Π _θ of ∂S _θ at x = x ₀, with respect to an inner normal, is non-decreasing in θ. Clearly (2.3) is equivalent to Π _θ ≤ Π₁ and the general case follows by replacing p ₁ by p _θ′ for any θ′ ∈ (0, 1]. Note that Π _θ is well defined at x ₀ since ∂S _θ ∈ C ² in some neighbourhood N 0 of x ₀. By extending the segment [p ₀, p ₁] beyond p ₀, we also have that Π _θ is bounded independently of θ. From (2.3), following the optimal transportation case in [2], it also follows that condition A3w can be expressed in terms of g and g _x only, using just condition A1, namely:

• A3v: For any ( x 0 , u 0 , [ p 0 , p 1 ] ) ⊂ U , we have

for x ∈ Ω₀, for any θ ∈ (0, 1).

Note that the “o” term in condition A3v may depend on θ.

In this section, we will prove the following further equivalent characterisations of condition A3w, when condition A2 and the dual condition A1* are also satisfied. As well as providing the relationship between condition A3w and the local g-convexity of sections S _θ , the result also strengthens condition A3v by removing the “o” dependence.

We remark here that the proof of Theorem 2.1 is somewhat different from that of the corresponding results in the optimal transportation case in Theorem 1.2 of [2] in that it avoids the measure theoretic argument in Lemma 2.3 of [2]. When the third derivatives g _xxy and g _xxz exist, so that the matrix function A is differentiable with respect to p and u, implication (i) follows directly from Lemma 2.4 in [1], in accordance with the optimal transportation case in Section 2.1 of [2]. Alternatively, in this case the mapping Q will be twice differentiable in x and we can simplify the proof of implication (i) through a C ² coordinate change to express h ₀ as an affine function in the q variable so that the result then follows straight from the monotonicity of Π _θ . From the boundedness of Π _θ , we at least have as in [2], that the g-hyperplane H ₀ is C ^1,1 smooth near x ₀ so that in the q coordinates we obtain the existence of a local supporting hyperplane almost everywhere near x ₀ and may then use the continuity of E, which implies H ₀ is C ¹ smooth, to obtain the local g-convexity in implication (i). We also note that here the equivalence of A3w and A3w(2) for C ⁴ cost functions in optimal transportation goes back to [7], where it is proved by approximation from the global regularity in [6] and also plays a fundamental role in showing the sharpness of condition A3w for regularity.

By modification of the preceding arguments we can prove analogous equivalent versions of the strong condition A3s, including its invariance under duality. For this it is convenient to fix a subset U ′ ⊂ ⊂ U . Then we can write condition A3s in the form

for any ( x 0 , u 0 , [ p 0 , p 1 ] ) ⊂ U ′ , 0 ≤ θ ≤ 1, ξ·(p ₁ − p ₀) = 0 and some positive constant δ, depending on U ′ . In place of (2.3), we then have, for h θ , δ ≔ h θ − δ 2 | p θ − p 0 | 2 | x − x 0 | 2 ,

for all ξ ∈ R n such that ξ·Dh _θ,δ (x ₀) = 0. Now, setting S _θ,δ = {x ∈ Ω₀|h _θ,δ (x) < 0}, we can state the following strong version of Theorem 2.1.

The case A3s ⇒ A3s(2) in Theorem 2.2 extends to C ² generating functions the corresponding result in Lemma 4.5 in [8], which is proved there, similarly to the basic convexity results under A3w, by using the differential inequality approach. We may also express the condition A3s(1) in terms of a local uniform g-convexity of S ₁. Note also that, without assuming the dual condition A1*, we obtain from the proof of the implication A3s ⇒ A3s(1), (in particular from the estimate (2.8) applied to h _θ,δ ), that condition A3s is equivalent to a strong form of condition A3v, corresponding to (2.9), namely

for all ( x 0 , u 0 , [ p 0 , p 1 ] ) ⊂ U ′ ⊂ ⊂ U , x ∈ Ω₀ and some positive constant δ, with the “o” dependence independent of θ.

3 Global convexity

In this section we deduce from Theorem 1.1, fundamental properties of g-convex functions when g is only assumed C ². First we recall from [1] that a function u ∈ C ⁰(Ω) is called g-convex in Ω, if for each x ₀ ∈ Ω, there exists ( y 0 , z 0 ) ∈ R n × R such that (Ω, y ₀, z ₀) ⊂ Γ and

for all x ∈ Ω. If u is differentiable at x ₀, then y ₀ = Tu(x ₀): = Y(x ₀, u(x ₀), Du(x ₀)), while if u is twice differentiable at x ₀, then

We also refer to functions of the form g(⋅, y ₀, z ₀) as g -affine and as a g -support at x ₀ in Ω if (3.1) is satisfied. Note also that the g-convexity of a function u in Ω implies its local semi-convexity.

If u is a g-convex function on Ω, extending the differentiable case, we define the g -normal mapping of u at x ₀ ∈ Ω to be the set:

where z ₀ = g*(x ₀, y ₀, u ₀), u ₀ = u(x ₀). Note that if u = g(⋅, y, z) is g-affine, then Tu = y, while in general

where ∂u denotes the sub differential of u, provided the extended one jet, J 1 [ u ] ( x 0 ) = [ x 0 , u ( x 0 ) , ∂ u ( x 0 ) ] ⊂ U

Next if g ₀ = g(⋅, y ₀, z ₀) is a g-affine function, we define the section of a g-convex function u with respect to g ₀ by

We can also have a notion of closed sections, (as used in [2]), given by

which includes, as a special case, the contact set of u with respect to g ₀,

when g ₀ is a g-support of u.

Our approach here will be based on the following global extension of Theorem 2.1(i), which extends result (ii) in Theorem 1.2 in [2] to the generating function case.

For further results we will use the sub-convexity notion introduced in Section 2 of [8] so that, for example, condition (ii) in Theorem 3.1 can be written equivalently as {y ₀, y} is sub g*-convex with respect to g ₀ on Ω for all y ∈ Tu(Ω) or that the g*-segment, with respect to (x, g ₀(x)), joining y ₀ and y is well defined for all x ∈ Ω, y ∈ Tu(Ω).

For convenience we recall here that a set S ⊂ R n is sub g-convex, with respect to (y ₀, z ₀), if (S, y ₀, z ₀) ⊂ Γ and the convex hull of Q 0 ( S ) ≔ Q ( ⋅ , y 0 , z 0 ) ( S ) ⊂ Q ( Γ ) . Analogously, a set S * ⊂ R n is sub g*-convex, with respect to (x ₀, u ₀), if x 0 , S * , u 0 ⊂ Γ * and the convex hull of P 0 ( S ) ≔ P ( x 0 , ⋅ , u 0 ) ( S ) ⊂ P ( Γ * ) . We then have that S is sub g-convex with respect to a function v : S * → R if S is sub g-convex with respect to each point on the graph of v and S* is sub g*-convex with respect to a function u : S → R if S* is sub g*-convex with respect to each point on the graph of u.

Recall also that the g-transform of a g-convex function u, on a domain Ω, is defined by

for y ∈ Tu(Ω), so that from (3.1), v(y ₀) = z ₀, if g ₀ = g(⋅, y ₀, z ₀) is a g-support to u.

From Theorem 3.1 and the invariance of A3w under duality, we then have the following global extension of Theorem 2.1(ii).

Note that by the semi-convexity of u, P(x ₀, u ₀, Σ₀) is the convex hull of P(x ₀, u ₀, Tu(x ₀)) so Corollary 3.1 follows from the g*- convexity of Tu(x ₀), which in turn follows directly using duality with domain Ω* a neighbourhood of Σ₀, which is also g*-convex with respect to x ₀, u ₀. But we may also proceed slightly differently as in [8] by proving first a special case when u is replaced by max{g ₀, g ₁} where g ₀ and g ₁ are two g-affine functions satisfying g ₁(x ₀) = g ₀(x ₀) = u ₀. Then using the notation in Theorem 3.2, if {x ₀, x} is sub g-convex with respect to (y _θ , z _θ ), for all θ ∈ (0, 1), we have the inequality,

which is the global version of A3w(2) in Theorem 2.1.

Finally we consider the global g-convexity of locally g-convex functions thereby extending Lemmas 2.1 in [1], [8] to the non-smooth case and part (iv) of Theorem 1.2 in [2] to the generating function case. Here we will define a function u ∈ C ⁰(Ω) to be locally g-convex in Ω, at any point x ₀ ∈ Ω, u has a g-support in some neighbourhood N 0 of x ₀.

Note that in [1], [8], we have defined local g-convexity for a C ² function u by the degenerate ellipticity condition (3.2). Clearly if u is elliptic in Ω, that is inequality (3.2) is strict in Ω, then u is locally g-convex as above. From this it follows by approximation, u → u + ϵ ( x − x 0 ) 2 , for small ϵ > 0 and Theorem 3.2, that our definitions are equivalent if A is Lipschitz continuous with respect to the u and p variables.

Corresponding to [2], the proof of Theorem 3.2 is just a modification of that of Theorem 3.1.

Finally we remark that from Theorem 2.2, by adapting the approach in [7], we can obtain the C ¹ and C ^1,α regularity of generalized solutions of the second boundary value for generated Jacobian equations for C ² generating functions satisfying A1,A2, A1* and A3s under appropriate integrability or boundedness conditions on the initial and target densities, f and f*, and convexity conditions on the initial and target domains, Ω and Ω*. In particular, the corresponding regularity results in the optimal transportation case, in Theorems 3.4 and 3.7 of [7], may be extended to C ² cost functions while their extensions to generated Jacobian equations, proved recently in Theorem 2.14 of [9], may be extended to C ² generating functions. In fact these extensions do not need the full strength of the implication A3s → A3s(2) in Theorem 2.2 and the estimate (2.16), which already is a refinement of Proposition 5.1 in [7] and Lemma 3.3 in [9], is sufficient. For this we also need the characterisation of the g-normal mapping in Corollary 3.1.

For a formulation of the generalized second boundary value problem for generated Jacobian equations we may refer, for example, to Section 4 in [1] or Section 3 in [8].