  We present an optimization problem emerging from optimal control theory and
situated at the intersection of fractional programming and linear max-min
programming on polytopes. A na\"ive solution would require solving four nested,
possibly nonlinear, optimization problems. Instead, relying on numerous
geometric arguments we determine an analytical solution to this problem. In the
course of proving our main theorem we also establish another optimization
result stating that the minimum of a specific minimax optimization is located
at a vertex of the constraint set.
