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

It is shown that for any positive integer n and any function in Lp([0, 1]) with p ∈ [1,∞) there exists a best approximation by linear combinations of n characteristic functions of half-spaces. Further, sequences of such linear combinations converging in distance to the best approximation distance have subsequences converging to the best approximation, i.e., these linear combinations are an approximatively compact set.