Characterizing best isotone approximations in LpLp spaces, 1≤p<∞1≤p<∞

We characterize best approximations in Lp(1≤p<∞) from a class of convex subcones of the convex cone of nonnegative functions and the convex cone of increasing functions; in particular, from the convex cones themselves. These subcones arise naturally in applications of constrained or shape-preserving approximation.