A new approach to the fuzzification of convex structures is introduced. It is also called anM-fuzzifying convex structure. In the definition ofM-fuzzifying convex structure, each subset can be regarded as a convex set to some degree. AnM-fuzzifying convex structure can be characterized by means of itsM-fuzzifying closure operator. AnM-fuzzifying convex structure and itsM-fuzzifying closure operator are one-to-one corresponding. The concepts ofM-fuzzifying convexity preserving functions, substructures, disjoint sums, bases, subbases, joins, product, and quotient structures are presented and their fundamental properties are obtained inM-fuzzifying convex structure.