Some Geometric Properties Related to Uniform Convexity of Banach Spaces

Abstract

In 1955, two generalizations of uniformly convex (UR) Banach spaces were introduced. Lovaglia [L] studied the locally uniformly convex (LUR) Banach spaces. Fan and Glicksberg [FG1,2] extended the 2R Banach spaces, introduced by V. Smulian, and studied the fully k-convex (kR) Banach spaces. Later, two more generalizations of uniform convexity were introduced for Banach spaces. In 1979, Sullivan [S] studied the k-uniformly rotund (k-UR) Banach spaces and in 1980, Huff [H] studied the nearly uniformly convex (NUC) Banach spaces. It is known that every k—UR space is NUC [Y] and every strictly convex k-UR space is (k-f 1)R [LY]. Recently, the locally fully k-convex (LkR) Banach spaces are defined in [NW] and it is proved that LUR L2R =*···=» LkR L(k+1)R and every strictly convex locally k-uniformly rotund Banach space is L(k+1)R. Furthermore, Kutzarova [K] introduces k—B Banach spaces and k—nearly uniformly convex (k—NUC) Banach spaces and shows that these are two classes of Banach spaces that lie strictly in between the classes of k-UR and kR spaces. The relationships between k/? and k-NUC spaces are also determined in [K]. The class of locally k—B and locally k-NUC Banach spaces are identical and has been studied in [KL]. By definitions, it follows that the NUC space is exactly the limit space of k-NUC when k —»<d. In this paper, we study the corresponding spaces of k—UR, Lk—UR, kR, LkR 282and L k-NUC when k —» od. The authors wish to thank Denka Kutzarova for sending them the preprint of her paper [K].