  The classical Pfaff-Darboux theorem, which provides local 'normal forms' for
$1$-forms on manifolds, has applications in the theory of certain economic
models [Chiappori P.-A., Ekeland I., Found. Trends Microecon. 5 (2009), 1-151].
However, the normal forms needed in these models often come with an additional
requirement of some type of convexity, which is not provided by the classical
proofs of the Pfaff-Darboux theorem. (The appropriate notion of 'convexity' is
a feature of the economic model. In the simplest case, when the economic model
is formulated in a domain in $\mathbb{R}^n$, convexity has its usual meaning.)
In [Methods Appl. Anal. 9 (2002), 329-344], Ekeland and Nirenberg were able to
characterize necessary and sufficient conditions for a given 1-form $\omega$ to
admit a convex local normal form (and to show that some earlier attempts
[Chiappori P.-A., Ekeland I., Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4 25 (1997),
287-297] and [Zakalyukin V.M., C. R. Acad. Sci. Paris S\'er. I Math. 327
(1998), 633-638] at this characterization had been unsuccessful). In this
article, after providing some necessary background, I prove a strengthened and
generalized convex Pfaff-Darboux theorem, one that covers the case of a
Legendrian foliation in which the notion of convexity is defined in terms of a
torsion-free affine connection on the underlying manifold. (The main result of
Ekeland and Nirenberg concerns the case in which the affine connection is
flat.)
