Cotype-Cotype Problem

The cotype-cotype problem asks whether an infinite-dimensional Banach space with the bounded approximation property must be K-convex when both it and its dual normed space have finite Rademacher cotype (Gupta et al. 2025, Conjecture 1.5). The finite cotype exponents for the two spaces need not agree. K-convexity implies these two finite-cotype conditions without an approximation property assumption.

The approximation property means approximation by finite-rank operators uniformly on compact sets, without a uniform bound on their operator norms. The bounded approximation property additionally requires one uniform finite bound on these operator norms.