Let $x_0$ be a point contained inside a compact, convex set $C\subset\mathbb{R}^d$.  Is there a computationally tractable way to find the largest ball, centered at $x_0$, that is contained inside $C$?  When $C$ is a polyhedron defined by linear inequalities, this problem is trivial (just check the minimum distance to the boundary of each of the half-spaces defined by the inequalities).