  This article examines differentiability properties of the value function of
positioning choice problems, a class of optimisation problems in
finite-dimensional Euclidean spaces. We show that positioning choice problems'
value function is always almost everywhere differentiable even when the
objective function is discontinuous. To obtain this result we first show that
the Dini superdifferential is always well-defined for the maxima of positioning
choice problems. This last property allows to state first-order necessary
conditions in terms of Dini supergradients. We then prove our main result,
which is an ad-hoc envelope theorem for positioning choice problems. Lastly,
after discussing necessity of some key assumptions, we conjecture that similar
theorems might hold in other spaces as well.
