Weakly Monotone Sets and Continuous Selection from a Near-Best Approximation Operator

Abstract

A new notion of weak monotonicity of sets is introduced, and it is shown that an approximatively compact and weakly monotone connected (weakly Menger-connected) set in a Banach space admits a continuous additive (multiplicative) ε-selection for any ε < 0. Then a notion of weak monotone connectedness (weak Menger connectedness) of sets with respect to a set of d-defining functionals is introduced. For such sets, continuous (d⁻¹, ε)-selections are constructed on arbitrary compact sets.