John-type decompositions for affinely-optimal positions of convex bodies

Many classical problems in convex geometry can be cast as optimization problems under certain containment conditions. The arguably best-understood example is volume-maximization of convex bodies contained in other convex bodies, where the John decomposition describes$\unicode{x2014}$and in the Euclidean case fully characterizes$\unicode{x2014}$the optimal positions. For many other such problems, however, no general optimality conditions are known. To address this, we generalize an approach of O. B. Ader to obtain a John-type decomposition as a necessary condition for affinely-optimal containment chains, i.e., chains $r L_1 + c \subseteq K \subseteq R L_2 + d$ for convex bodies $K, L_1, L_2 \subseteq \mathbb{R}^n$, translation vectors $c,d \in \mathbb{R}^n$, and reals $r,R > 0$ such that the ratio $\frac{R}{r}$ cannot be decreased by linearly transforming $K$. We again obtain sufficiency for optimality when ellipsoids are involved, and show how optimality conditions for various problems follow from our result. Our main applications concern the Banach-Mazur distance, where we provide necessary optimality conditions in the general case and a full characterization in the Euclidean case. Finally, we derive several consequences of these optimality conditions related to the Banach-Mazur distance to the Euclidean ball.