Almost Chebyshev set with respect to bounded subsets

The uniqueness and existence of restricted Chebyshev center with respect to arbitrary subset are investigated. The concept of almost Chebyshev sets with respect to bounded subsets is introduced. It is proved that each closed subset in a reflexive locally uniformly convex (uniformly convex, respectively) Banach space is an almost Chebyshev subset with respect to compact convex subsets (bounded convex subsets and bounded subsets, respectively). Project supported by the National Natural Science Foundation of China, Natural Science Foundation of Zhejiang Province, and the State Major Key Project for Basic Researchers of China.