Moduli of uniform convexity for convex sets

Let C be a proper, closed subset with nonempty interior in a normed space X. We define four variants of modulus of convexity for C and prove that they all coincide. This result, which is classical and wellknown for C = B_X (the unit ball of X), requires a less easy proof than the particular case of B_X. We also show that if the modulus of convexity of C is not identically null, then C is bounded. This extends a result by M.V. Balashov and D. Repovs.