Existence of best N-convex approximants in L1

We consider here best approximation by n-convex functions. We first show that if f∈L₁0,1], then there is, a best L₁-approximant to f by functions which are n-convex on (0,1). We then show that if f∈L_∞0,1], then any best L_p-approximant, f_p, to f by n-convex, functions is bounded and hence, f has the Pólya-one property, i.e., f_p converges a.e. as p decreases to one.