Weak-star dentability, quasi-weak-star near dentability and continuity of metric projector in Banach spaces

. The relations between the w ∗ dentability and Chebyshev set or the continuity of metric projection operator are given. Let X ∗ be the conjugate space of Banach space X , the conditions of a point ( x ∗ , y ∗ ) on the unit sphere of product space X ∗ × X ∗ to be w ∗ denting point of closed unit ball of product space X ∗ × X ∗ are given. Also, a notion of quasi-w ∗ near dentability in conjugate space X ∗ is introduced and the relations between the quasi-w ∗ nearly denting point of closed unit ball of X ∗ and a certain slice of closed unit ball of X ∗ are given.