What Is Spacetime Invexity? Time-Dependent and Variational Extensions of Hanson Invexity

Hanson introduced invexity in 1981, and Ben-Israel and Mond answered “What is in-vexity?” in 1986: for a differentiable function, invexity is equivalent to every stationarypoint being a global minimum. We ask the analogous question for problems that de-pend on time, where the objective is a functional J(x) =∫︁ T0 f (t, x, ˙x) dt of a path.Here “spacetime” means the time and state variables (t, x); no relativistic structure isused. Three points describe the answer:• When the integrand does not involve ˙x, the problem decouples in time and theclassical characterization applies at each instant;• With derivative coupling, a variational invexity condition, with a kernel thatvanishes at the fixed endpoints, makes the Euler–Lagrange equation sufficient fora global minimum;• A worked example has a Lagrangian that is not jointly convex, yet its extremalis the global minimizer (value 4, against 4.8 for the straight line).A dictionary of the three ingredients, When (time), Where (the kernel, or direction ofinvexity) and What (the functional), is given. Variational invexity is known in the lit-erature; the contribution is an exposition, the dictionary and the example. Constraintsare not treated.