Extreme Points of Convex Sets

Given a nonempty set $E \subset \R^n$, we provide necessary and sufficient conditions for the existence of a convex set $K \subset \R^n$ (possibly, nonclosed and unbounded) such that $\Ext K = E$. Also, we describe a family of convex sets $K \subset \R^n$ satisfying the equality $K = \Conv (\Ext K)$, and, more general, $K = \Conv (\Ext K) + \Rec K$, where $\Rec K$ denotes the recession cone of $K$.