Liu–Pego's local convexity conjecture for piecewise affine Lipschitz maps

Let $\Omega \subset \mathbb R^d$ be an open set, and let $\{A_i\}$ be a partition of $\Omega$ up to a negligible set, with each $A_i$ open. Let $\phi$ be a Lipschitz map that is affine on every $A_i$, and set

$$F=\{x\in\Omega\mid \phi\text{ is differentiable at }x\}.$$

Assume that, for almost every $t>0$, the map $x\mapsto x+t\nabla\phi(x)$ is injective on $F$.

Liu–Pego's local convexity conjecture. Every such map $\phi$ must necessarily be locally convex in $\Omega$.

The conjecture asks whether almost-everywhere injectivity of the maps generated by the gradient forces local convexity. The paper proves the result under the additional assumption $\nabla\phi\in BV_{\mathrm{loc}}(\Omega)$; without this assumption, the problem is described as almost completely open.