The cotype-cotype conjecture: finite cotype of a Banach space and its dual implies K-convexity (approximation property case)

Theorem 1.1: a nonzero real Banach space $X$ with the approximation property is $K$-convex if and only if $C_q(X)<\infty$ and $C_r(X^*)<\infty$ for some $q,r\in[2,\infty)$, possibly different; equivalently such $X$ has nontrivial Rademacher type. Corollary 1.2 makes the bound on $K(X)$ uniform in the cotype data and gives dimension-free metric-entropy duality for the corresponding convex bodies; a further consequence settles a Gupta-Misra-Ray tensor-norm conjecture (their Conjecture 1.4) under AP. Real scalars only; constants are existential. It does not address spaces failing AP, where the statement is false.

Theorem 1.1: a nonzero real Banach space $X$ with the approximation property is $K$-convex if and only if $C_q(X)<\infty$ and $C_r(X^*)<\infty$ for some $q,r\in[2,\infty)$, possibly different; equivalently such $X$ has nontrivial Rademacher type. Corollary 1.2 makes the bound on $K(X)$ uniform in the cotype data and gives dimension-free metric-entropy duality for the corresponding convex bodies; a further consequence settles a Gupta-Misra-Ray tensor-norm conjecture (their Conjecture 1.4) under AP. Real scalars only; constants are existential. It does not address spaces failing AP, where the statement is false.