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K-Convex Space

A K-convex space is a Banach space X on which the Rademacher projections on finite sign cubes are uniformly bounded. For f:{-1,1}^s->X, with uniform probability on the cube, define

R_sf(epsilon)=sum_(j=1)^sepsilon_jE_eta[eta_jf(eta)].

The condition is sup_(s>=1)||R_s||_(L^2(X)->L^2(X))<infty. This is a uniform condition over all finite numbers of signs. It is related to the Rademacher cotype of both X and its dual normed space through the cotype-cotype problem.

See also

Banach Space, Cotype-Cotype Problem, Dual Normed Space, Rademacher Cotype

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References

Gupta, R.; Misra, G.; and Ray, S. K. "On Variants of the Grothendieck Inequality and Estimates on Tensor Product Norms." 12 Aug 2025. https://arxiv.org/abs/2305.13270v2.Pisier, G. "Semi-groupes holomorphes et K-convexité." Sém. Anal. Fonct. École Polytechnique, Exposé 2, 1-30, 1980-1981. https://www.numdam.org/item/SAF_1980-1981____A2_0/.

Cite this as:

Weisstein, Eric W. "K-Convex Space." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/K-ConvexSpace.html

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