I have this simple question related to convex set and convex polyhedron. As the content in the title, it's basically my question:

Is closed convex set with finite number of extreme points convex polyhedron

If not, can you please give me some hint to find the counter-example of that statement? Thanks a lot. I really appreciate

  • $\begingroup$ What is a polyhedron? $\endgroup$ – gerw Jan 23 '16 at 7:19
  • $\begingroup$ Sorry, my bad. It should be convex polyhedron. In wikipedia, "convex polyhedron is any point set that is the intersection of a finite number of half-spaces" $\endgroup$ – le duc quang Jan 23 '16 at 7:24

You additionally need that your set is bounded, otherwise it may have too few extreme points:

  • any convex, closed cone has only one extreme point
  • For any closed, convex $C$, consider $C \times \{0\}$: no extreme points.

If your set is bounded, it is (assuming that the ambient space is finite-dimensional) compact. By Krein-Milman, the set is the convex hull of its extreme points, hence, a polyhedron.

  • $\begingroup$ Of course "convex hull of a finite set" includes things that it seems to me would at best be described as degenerate polyhedra; for example do we really want to call a line segment in $\Bbb R^3$ a polyhedron? {Of course}^2 the definition "intersection of finitely many (closed) half-spaces" also includes degenerate polyhedra (for a few seconds I thought that that segment was not the intersection of finitely many half spaces, but of course it is...) $\endgroup$ – David C. Ullrich Jan 23 '16 at 13:08
  • $\begingroup$ @DavidC.Ullrich: Polytopes are defined as convex hull of a finite set of points - and the only difference is that polyhedrons might be unbounded. And both definitions include "degenerate" sets. In fact, a line segment in $\mathbb{R}^3$ is a non-degenerate one-dimensional polyhedron, that is embedded in a three-dimensional space. $\endgroup$ – gerw Jan 23 '16 at 13:32
  • $\begingroup$ Well fine then. You're saying I need to know the definitions, eh? I'll have to think about that. interesting concept, heh... $\endgroup$ – David C. Ullrich Jan 23 '16 at 13:46
  • $\begingroup$ People wrote whole book about convex polytopes, so it seems to be an interesting topic :) $\endgroup$ – gerw Jan 23 '16 at 14:17
  • $\begingroup$ @DavidC.Ullrich: Can you suggest some good books about those topics. They are really interesting to me. Thanks a lot. $\endgroup$ – le duc quang Jan 30 '16 at 5:03

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