A minimizing property of best monotone approximants on their level surfaces

In this paper we show that a best monotone ϕ-approximant g on the cube (-1,1)n to a continuous and bounded function f is a best monotone ϕ-approximant on the level surfaces of g with respect to certain n-1-dimensional measure. This measure is mutually absolutely continuous with respect to the n-1-dimensional Haussdorff measure.