Existence of Best n-Convex Approximations in L1

We consider here best approximation by n-convex functions. We first show that if f∈L1[0,1], 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 Lp-approximant, fp, to f by n-convex, functions is bounded and hence, f has the Pólya-one property, i.e., fp converges a.e. as p decreases to one