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

Let

be a finite set in a real linear space and let

{\mathcal J}_S J S

be a family consisting of

|S| ∣ S ∣

intervals in

\mathbb{R} R

. In this paper we deal with a convex operator

\operatorname{co}(S,{\mathcal J}_S) co ⁡ ( S , J S )

called the convex interval hull. This operator generalizes the familiar concepts of the convex hull,

\operatorname{conv}(S) conv ⁡ ( S )

, and the affine hull,

\operatorname{aff}(S) aff ⁡ ( S )

, of

S S

. The set

\operatorname{co}(S,{\mathcal J}_S) co ⁡ ( S , J S )

is a convex subset of the linear space and can be either bounded or unbounded, depending on the families

{\mathcal J}_S J S

. In this paper we apply

\operatorname{co}(S,{\mathcal J}_S) co ⁡ ( S , J S )

to obtain unbounded images of a finite set

S S

. As special images of

\operatorname{co}(S,{\mathcal J}_S) co ⁡ ( S , J S )

for finite

S S

we obtain such unbounded objects as: hyperplanes, cylinders, cones, penumbras and wedges. We also apply

\operatorname{co}(S,{\mathcal J}_S) co ⁡ ( S , J S )

to study some properties of extreme points. In relation to

\operatorname{co}(S,{\mathcal J}_S) co ⁡ ( S , J S )

we introduce the so-called extreme interval operator

\operatorname{Eco}(S) Eco ⁡ ( S )

and prove some analogues of the celebrated Minkowski-Krein-Milman's theorem.