Historically, problems of simultaneous inner and outer approximation under affinity – such as bounding the Banach-Mazur distance – have often been approached indirectly by relying on extremal configurations for other optimization problems, with the hope that they behave well for the problem at hand. Prominent examples include volume-extremal approximations like the John and Loewner ellipsoids, or the isotropic position, used recently by Bizeul and Klartag to obtain a near-optimal bound on the Banach-Mazur distance between general convex bodies. While such approaches have led to several tight results and have been influential even in other branches of mathematics, they do not actually consider the problem of finding optimal simultaneous inner and outer approximations under affine transformations. We address this gap by establishing necessary optimality conditions for such simultaneous approximations. For approximation by ellipsoids, these conditions fully characterize the optimal solutions. We also discuss applications, including proofs of the upper bounds on the Banach-Mazur distance to the Euclidean ball that allow a more direct analysis of the extremal cases.