Polynomial estimates for K-convexity from Walsh decay and cotype

We obtain polynomial bounds for the norm K(X) of the Rademacher projection on a Banach space X. If the probability-normalized Walsh transform on the s-dimensional cube has norm at most 1/2 on L2(X), then K(X) ≤ 9000s log(6000s2). Consequently, for each 1 < p ≤ 2, the type constant Tp(X) satisfies K(X) ≲p Tp(X)p/(p−1)(1 + log Tp(X)). This improves the exponential upper bound in Naor’s 2022 quantitative question, without determining optimal dependence.

We also quantify the cotype–cotype theorem under the approximation property. For finite q, r ≥ 2, put t = max{q, r} and H = Cq(X)Cr(X*), where Cq and Cr are Rademacher cotype constants. If X has the approximation property and H < ∞, then K(X) ≤ AtH6t(1 + log H), with an explicit constant depending only on t. The analytic step uses the coordinate structure of cube noise to turn a sector bound into a disk bound with polynomial loss. In the cotype argument, deleting small Fourier coefficients makes a previously qualitative coefficient estimate effective.

GitHub repository: https://github.com/eulogica/polynomial-k-convexity-walsh-cotype