  The widespread application of modern machine learning has increased the need
for robust statistical algorithms. This work studies one such fundamental
statistical measure known as the Tukey depth. We study the problem in the
continuum (population) limit. In particular, we derive the associated necessary
conditions, which take the form of a first-order partial differential equation.
We discuss the classical interpretation of this necessary condition as the
viscosity solution of a Hamilton-Jacobi equation, but with a non-classical
Hamiltonian with discontinuous dependence on the gradient at zero. We prove
that this equation possesses a unique viscosity solution and that this solution
always bounds the Tukey depth from below. In certain cases, we prove that the
Tukey depth is equal to the viscosity solution, and we give some illustrations
of standard numerical methods from the optimal control community which deal
directly with the partial differential equation. We conclude by outlining
several promising research directions both in terms of new numerical algorithms
and theoretical challenges.
