  We introduce the moduli of the supporting convexity and the supporting
smoothness of a Banach space, which characterize the deviation of the unit
sphere from an arbitrary supporting hyperplane. We show that the modulus of
supporting smoothness, the Bana{\'s} modulus, and the modulus of smoothness are
all equivalent at zero, the modulus of supporting convexity is equivalent at
zero to the modulus of convexity. We prove a Day--Nordlander type result for
these moduli.
