Timeline for Polytopes whose intersections have few vertices
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Mar 26, 2019 at 9:23 | comment | added | Guillaume Aubrun | You can realize a combinatorial hypercube as the intersection of two simplices ; this is an example with exponentially many vertices | |
| Apr 13, 2017 at 12:57 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Apr 3, 2015 at 23:58 | comment | added | Richard Stanley | An intersection of two simplices in $\mathbb{R}^n$ has at most $2n+2$ facets. By the Upper Bound Theorem for polytopes, the number of vertices is at most $2\sum_{i=0}^{(n-1)/2}{n+i+1\choose i}$ if $n$ is odd, and $2\sum_{i=0}^{\frac n2-1}{n+i+1\choose i} + {\frac{3n}{2}+1\choose \frac n2}$ if $n$ is even. I don't know how close one can come to achieving these bounds. | |
| Apr 3, 2015 at 1:47 | comment | added | Richard Stanley | @JenniferGao: no problem. It's a nice question. | |
| Apr 3, 2015 at 1:16 | comment | added | Joseph O'Rourke | You question suggests this (easier) one: What is the maximum number of vertices of an intersection of two simplices in $\mathbb{R}^n$? | |
| Apr 2, 2015 at 22:31 | comment | added | Gerhard Paseman | Welcome to MathOverflow! Perhaps they will make a "Human" badge for "edit on first post". Gerhard "Has Plenty Of Badges Already" Paseman, 2015.04.02 | |
| Apr 2, 2015 at 22:28 | history | edited | Jennifer Gao | CC BY-SA 3.0 |
added 18 characters in body
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| Apr 2, 2015 at 22:28 | comment | added | Jennifer Gao | Meant to say "unit ball in $l_1$", how embarassing... | |
| Apr 2, 2015 at 22:19 | comment | added | Richard Stanley | The unit ball is not a polytope. | |
| Apr 2, 2015 at 19:42 | history | asked | Jennifer Gao | CC BY-SA 3.0 |