Local characterization of strongly convex sets

Strongly convex sets in Hilbert spaces are characterized by local properties. One quantity which is used for this purpose is a generalization of the modulus of convexity δΩδΩ of a set ΩΩ. We also show that limε→0δΩ(ε)/ε2limε→0δΩ(ε)/ε2 exists whenever ΩΩ is closed and convex.