The Distance between a Symmetric Space and a 2‐Convex or 2‐Concave Space

If X is a k ‐dimensional symmetric space and p is a real number with 1 ⩽ p ⩽ ∞ then the Banach‐Mazur distance d ( X ,l p k ) ⩽ c √ k , where c is a universal constant. The estimate remains valid if one replacesl p kby a k ‐dimensional space Y with unconditional basis which is either 2‐convex or 2‐concave. If both X and Y have unconditional bases and Y is either 2‐convex or 2‐concave, then d ( X, Y ) ⩽k ( 1 + log k ).