Convex Interval Hull of Finite Sets in Real Linear Spaces: Extreme Points and Unbounded Images

Let $S$ be a finite set in a real linear space and let $\cJ_S$ be a family consisting of $|S|$ intervals in $\rr$. In this paper we deal with a convex operator $\co(S,\cJ_S)$ called the convex interval hull. This operator generalizes the familiar concepts of the convex hull, $\conv(S)$, and the affine hull, $\aff(S)$, of $S$. The set $\co(S,\cJ_S)$ is a convex subset of the linear space and can be either bounded or unbounded, depending on the families $\cJ_S$. In this paper we apply $\co(S,\cJ_S)$ to obtain unbounded images of a finite set $S$. As special images of $\co(S,\cJ_S)$ for finite $S$ we obtain such unbounded objects as: hyperplanes, cylinders, cones, penumbras and wedges. We also apply $\co(S,\cJ_S)$ to study some properties of extreme points. In relation to $\co(S,\cJ_S)$ we introduce the so-called extreme interval operator $\Eco(S)$ and prove some analogues of the celebrated Minkowski-Krein-Milman's theorem.