The cornerstone of Asymptotic Geometric Analysis is the celebrated theorem of Dvoretzky on almost-spherical sections of high-dimensional convex bodies. The critical dimension at which spherical sections appear, on an isomorphic scale, was settled by V. Milman's seminal work in the 70's. On the other hand, the dimension on the almost-isometric scale (in the existential or in the random case) is far from being understood.  Motivated by questions arising in the study of this problem, we are going to discuss a variance-sensitive concentration inequality for convex functions of Gaussian vectors. Time permitting we will discuss recent developments on Dvoretzky's theorem for special classes of bodies. The talk will be based on joint works with Petros Valettas.