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I am currently doing research in generalizing the concept of a convex kernel. I would appreciate any links or resources pertaining to current research and basic results about convex kernels.

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Perhaps you already have: Sparks, Characterization of the Generalized Convex Kernel, Proceedings of the American Mathematical Society, Vol. 27, No. 3, Mar., 1971, 563-65. It's old but a good jumping-off point and you could check subsequent references to it?