Take a piecewise affine Lipschitz function φ on a domain in ℝᵈ. Consider the map x ↦ x + t∇φ(x) — a perturbation of identity along the gradient. Require that this map is injective for almost every t > 0: distinct points stay distinct under the flow.
Bianchini and Talamini prove that φ must be locally convex, provided its gradient has bounded total variation. The injectivity requirement — a global, qualitative condition about the map not collapsing distinct points — forces a local, quantitative condition about the function's curvature.
The result resolves a conjecture of Liu and Pego from 2025. The mechanism: if φ were not locally convex, the gradient flow would develop self-intersections at some positive time. Two initially separated trajectories would collide because non-convexity creates converging gradient directions. The bounded total variation of the gradient prevents oscillation from hiding these collisions — the gradient cannot fluctuate so wildly that collisions exist but on a measure-zero set of times.
The structural insight: injectivity is a topological property (no point has two preimages). Convexity is a geometric property (the Hessian is positive semidefinite). The theorem says that for gradient perturbations of the identity with BV regularity, topology determines geometry. You cannot have a non-convex function whose gradient flow preserves distinctness — the non-convexity inevitably produces merging trajectories, and bounded variation ensures the merging happens on a set of positive measure in t.
The extension to general measure-preserving maps broadens the result: the conclusion holds not just for identity perturbations but for any volume-preserving transformation under the same regularity assumptions.