Positioning choice problems and their applications to the theory of moral hazard

I study a class of optimisation problems in finite-dimensional Euclidean spaces for which the value function is almost everywhere differentiable even when objective function is discontinuous. I call this class of problems positioning choice problem as it has a straightforward geometrical interpretation as a choice of position. I collect observations about this class of problems as a collection of "Ad-Hoc" envelope theorems for discontinuous objective functions. I apply the findings to the theory of moral hazard / mechanism design. In particular, I study models of fraud in the design of securities and insurance.