Relationships between the oriented distance functional and a nonlinear separation functional

In this paper, we study two nonlinear scalarization functionals, namely, the oriented distance functional and a nonlinear separation functional due to Tammer. First, we prove the monotonicity of the oriented distance functional when the associated set is neither a cone nor a convex set. Then, we show several relationships between the two nonlinear scalarization functionals. By using these results, we also obtain an estimation of the sublevel set for the oriented distance functional.