In this paper, we focus on non-conservative collision avoidance between robots and obstacles with control affine dynamics and convex shapes. System safety is defined using the minimum distance between the safe regions associated with robots and obstacles. However, collision avoidance using the minimum distance as a control barrier function (CBF) can pose challenges because the minimum distance is implicitly defined by an optimization problem and thus nonsmooth in general. We identify a class of state-dependent convex sets, defined as strongly convex maps, for which the minimum distance is continuously differentiable, and the distance derivative can be computed using KKT solutions of the minimum distance problem. In particular, our formulation allows for ellipsoid-polytope collision avoidance and convex set algebraic operations on strongly convex maps. We show that the KKT solutions for strongly convex maps can be rapidly and accurately updated along state trajectories using a KKT solution ODE. Lastly, we propose a QP incorporating the CBF constraints and prove strong safety under minimal assumptions on the QP structure. We validate our approach in simulation on a quadrotor system navigating through an obstacle-filled corridor and demonstrate that CBF constraints can be enforced in real time for state-dependent convex sets without overapproximations.