Recent empirical research indicates that many convex optimization problems with random constraints exhibit a phase transition as the number of those constraints increases. In this talk, we will explicitly connect this phase transition with the asymptotic Gaussian fluctuations of the intrinsic volumes of the descent cone that is canonically attached to the convex optimization problem at hand. Our approach will be based on a variety of techniques, including (1) Steiner formulae for closed convex cones, (2) Stein's method and second order Poincaré inequality, (3) concentration estimates, and (4) Fourier analysis. This is a joint work with Larry Goldstein (Univ. of Southern California) and Giovanni Peccati (Univ. of Luxembourg).