Best Rational Approximation and Strict Quasi-Convexity

If a continuous function is strictly quasi-convex on a convex set $\Gamma $, then every local minimum of the function must be a global minimum. Furthermore, every local maximum of the function on the interior of $\Gamma $ must also be a global minimum. Here, we prove that any minimax rational approximation problem defines a strictly quasi-convex function with the property that a best approximation (if one exists) is a minimum of that function. The same result is not true in general for best rational approximation in other norms.