On some geometric properties concerning closed convex sets

In this article, we consider the (weak) drop property, weak property ($\alpha$), and property ($\omega$) for closed convex sets. Here we give some relations between those properties. Particularly, we prove that $C$ has (weak) property ($\alpha$) if and only if the subdifferential mapping of $C^0$ is ($n-n$) (respectively, ($n-w$)) upper semicontinuous and (weak)compact valued. This gives an extension of a theorem of Giles and the first author.