Best approximation in normed linear space is a well studied subject and it has numerous applications in many other streams in analysis. The theory have some nice interplay with the geometry of the unit ball and the differentiability of the dual norm. If the best approximation is guaranteed for a given subspace it is natural to study how this set of best approximation is located. Various strengthening of best approximation comes in this context. Instead of finding best approximation from a given point to a linear subspace one can generalize this notion for a given closed and bounded subset from the same linear subspace and hence the center of a arbitrary closed bounded set can be defined. Similar to the continuity of metric projection the continuity of chebyshev center is also of special interest. I will try to discuss about these topics, related well known results and some recent observations.
