Timeline for Nonzero convex combinations of convex hull vertices to yield an inner point
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 25, 2012 at 23:22 | comment | added | Sergei Ivanov | Any convex polytope is the convex hull of its vertices. A simplex contains many convex polytopes that are not simplices. For example, consider a quadrangle inside a triangle. | |
| Nov 25, 2012 at 23:15 | comment | added | 4fj | Answered my own question; it seems that's only the case when the vertices of the convex hull are affinely independent. | |
| Nov 25, 2012 at 22:44 | comment | added | 4fj | Yes, finite dimensions. What I'm trying to ask is: is the convex hull of a set of points in a finite dimensional probability simplex a Choquet simplex? Please excuse my lack of knowledge in these areas... it's funny where research can land you. | |
| Nov 25, 2012 at 22:30 | comment | added | Sergei Ivanov | Is this still about finite dimensions? If so, a Choquet simplex is just an ordinary simplex, otherwise you need to consider integrals rather than linear combinations. | |
| Nov 25, 2012 at 22:25 | comment | added | 4fj | Another question: would I be correct in saying that the convex hull of any set of points in a simplex is a Choquet simplex, which implies that in this case not only does such a nontrivial convex combination of convex hull vertices exist, but that the convex combination is unique? | |
| Nov 25, 2012 at 21:07 | vote | accept | 4fj | ||
| Nov 25, 2012 at 21:07 | comment | added | 4fj | That settles it! Any interior point of a convex hull can be expressed as a nontrivial convex combination of the hull vertices. | |
| Nov 25, 2012 at 20:58 | history | answered | Sergei Ivanov | CC BY-SA 3.0 |