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 well-known for C=BX C = B X (the unit ball of X ), requires a less easy proof than the particular case of BX. 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. Repovš.