Almost convexity and continuous selections of the set-valued metric generalized inverse in Banach spaces

In this paper, we define almost convex space. Let T : X -> Y be a linear bounded operator. This paper shows that: (1) If X is almost convex and 2-strictly convex, Y is a Banach space, D(T) is closed, N(T) is an approximatively compact Chebyshev subspace of D(T) and R(T) is a 2-Chebyshev hyperplane of Y, then there exists a homogeneous selection T-sigma of T-partial derivative such that continuous points of T-sigma is dense on Y. (2) If X is locally uniformly convex, Y is reflexive, D(T) is closed, N(T) is a proximinal subspace of D(T) and R(T) is a closed hyperplane of Y, then T-partial derivative is single-valued, homogeneous and continuous on Y. The results are a perfect answer to the open problem posed by Nashed and Votruba (Bull Am Math Soc 80:831-835, 1974).