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We introduce the moduli of the supporting convexity and the supportingsmoothness of a Banach space, which characterize the deviation of the unitsphere from an arbitrary supporting hyperplane. We show that the modulus ofsupporting smoothness, the Bana{\'s} modulus, and the modulus of smoothness areall equivalent at zero, the modulus of supporting convexity is equivalent atzero to the modulus of convexity. We prove a Day--Nordlander type result forthese moduli.

Modulus of supporting convexity and supporting smoothness