Best approximation by linear combinations of characteristic functions of half-spaces

It is shown that for any positive integer n and any function f in with p∈[1,∞) there exist n half-spaces such that f has a best approximation by a linear combination of their characteristic functions. Further, any sequence of linear combinations of n half-space characteristic functions converging in distance to the best approximation distance has a subsequence converging to a best approximation, i.e., the set of such n-fold linear combinations is an approximatively compact set.