Abstract

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

Author information

Contact details are reproduced from the original publication and may be historical.

Branko Ćurgus

Dept. of Mathematics, Western Washington University, Bellingham, U.S.A.

curgus@wwu.edu

Keywords

  • Convex set
  • cone
  • convex interval hull
  • extreme point
  • extreme interval operator
  • Minkowski-Krein-Milman property

Mathematics Subject Classification

52A05, 15A03, 14N20

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Suggested citation

B. Ćurgus, K. Kołodziejczyk. “Convex Interval Hull of Finite Sets in Real Linear Spaces: Extreme Points and Unbounded Images.” Journal of Convex Analysis 33 (2026), No. 1&2, 57–74. https://doi.org/10.68381/jca33004

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Published by Heldermann Verlag, 2026. Rights now held by Banach Press LLC.