Approximation in Semi-compact and Pre-concave Sets in Metric Spaces

Abstract

It is known that an approximation in non-compact sets is not guaranteed always same is true for concave sets which do not guarantee get the uniqness of a best approximation element. In this papers we introduce a new method to approximation in a semi-compact and pre-concave set in the metric spaces. This new method accomplished by prove some theorems which used properties of above set that guarantees existences and uniqueness of best approximation element for any element in this set, where these properties that is semi-compactness and pre-concavity alone are considered to be weak topological (metric) properties. Moreover, the other important conclusion is that the semi -compact and pre-concave set in metric space must be become a convex set, more than it must be a strictly convex set and the importance of this conclusion is clear that the convexity of sets is considered a strong properties of sets, also there are others conclusions in the folds of this research.