A note on δ-monotone maps in Euclidean spaces

We investigate in the Euclidean framework the behaviour of δ-monotone set-valued maps. This property arises as the natural extension to multivalued maps of the analogous notion enjoyed, for instance, by the gradient of the so-called quasiuniformly convex functions. While on any open subset of the domain the mere property of δ-monotonicity for multivalued maps entails both its single-valuedness and continuity, the behaviour at the boundary points of the domain is, definitely, unpredictable. We consider smooth points of the boundary of the domain, and we show that if a We investigate in the Euclidean framework the behaviour of δ-monotone set-valued maps. This property arises as the natural extension to multivalued maps of the analogous notion enjoyed, for instance, by the gradient of the so-called quasiuniformly convex functions. While on any open subset of the domain the mere property of δ-monotonicity for multivalued maps entails both its single-valuedness and continuity, the behaviour at the boundary points of the domain is, definitely, unpredictable. We consider smooth points of the boundary of the domain, and we show that if a δ-monotone map is upper semicontinuous, with convex and closed values, then it is single-valued also at those points.