The locally k-uniformly extremely convex and midpoint locally k-uniformly extremely convex Banach spaces

The locally k-uniformly extremely convex Banach spaces are introduced. It is shown here that every locally k-uniformly extremely convex space is locally (k+1)-uniformly extremely convex space, but the converse implication is not true. Some properties of locally k-uniformly extremely convex spaces are given. Specially, it is proved that if the second dual of Banach space X is locally 2-uniformly extremely convex, then X is reflexive. In addition, the notion of midpoint locally k-uniformly extreme convexity is introduced and its relations to other type convexity are given.