k-Extreme Convexity and k-Extreme Smoothness in Locally Convex Spaces

In this paper,the dual notions of k-extreme convexity and k-extreme smoothness are introduced in locally convex spaces,which are generalization of both k-extreme convexity(kextreme smoothness)in Banach spaces.The relationship between them and the other convexity(smoothness)are discussed.In addition,on the conditions of P-reflexivity,we obtain further the equivalent dual theorem of between k-extreme convexity and k-extreme smoothness.