On projections and monotony in Lp -spaces

In this paper we investigate the connection between the range of nearest point projections in L_p -spaces and monotony properties of the projection operator. We show e.g. that a nearest point projection onto a closed convex subset of an L_p -space (1) is monotone if and only if the closed convex range is a lattice. If the range is closed linear instead of closed convex then it turns out that positivity of the projection operator implies monotony, although the projection is in general not a linear operator. We can apply these results to a lot of known cases and to a case, in which the monotony of the projection operator was unknown up to now.