The promised geometric property of reflexive Banach spaces is the following: if "C" is a closed non-empty convex subset of the reflexive space "X", then for every "x" in "X" there exists a "c" in "C" such that minimizes the distance between "x" and points of "C". This follows from the preceding result for convex functions, applied to . Note that while the minimal distance between "x" and "C" is uniquely defined by "x", the point "c" is not. The closest point "c" is unique when "X" is uniformly convex.