The aim of this paper is to design regular feedback controls for the global asymptotic stabilization (GAS) of affine control systems with compact (convex) control value sets (CVS) U subset of R-m with 0 is an element of intU, in the framework of Artstein-Sontag's control Lyapunov function (CLF) approach. Convex analysis allows us to reveal the intrinsic geometry involved in the CLF stabilization problem, and to solve it, if an optimal control (omega) over bar (x) exists. The study of its existence, uniqueness, and continuity, in terms of properties of U, yields that (omega) over bar (x) is gradient-based, and leads to the generic class U(R-m) of compact strictly convex CVS. Moreover, higher regularity is attained via the geometry (curvature) of U (illustrated for the p, r-weighted balls). However, since (omega) over bar (x) is singular, we consider a general form of admissible feedback controls for the GAS of a system, provided a CLF is known. For U is an element of U(R-m), we design an explicit formula for suboptimal admissible controls, hence generically solving the synthesis problem entailed by Artstein's theorem. Finally, for a dense class of CVS, if we assume smoothness on the system and the CLF, we obtain an explicit formula for practically smooth feedback controls. The results are illustrated with the limit-cycle suppression of a system in R-3 via admissible controls.