Timeline for Graduate level convexity - Intersection of an r-polytope with a hyperplane is an r-1 polytope
Current License: CC BY-SA 4.0
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| Apr 25, 2022 at 13:09 | comment | added | Tryer | @Bma It is indeed true that every polytope can be represented as a convex hull and equivalently as the intersection of a finite number of closed halfspaces. Perhaps this is a good way to prove the result. Let me think about it. Thanks for your inputs. | |
| Apr 25, 2022 at 12:57 | comment | added | Bma | Letting $x_1,...,x_n$ be coordinates on $\mathbb{R}^n$, I believe a convex polytope of dimension $n$ is simply a compact region with nonempty interior defined by inequalities $0 \le f$ where $f$ is an affine function of the $x_i$ (please correct me if I am wrong on this definition). If you take the hyperplane to be $x_1 = 0$, it seems clear that these conditions are still satisfied by the intersection-- the intersection is obviously compact, has nonempty interior in $\mathbb{R}^{n-1}$ (unless it degenerates to a single point or an empty set), and is defined by affine inequalities. | |
| Apr 25, 2022 at 10:43 | history | asked | Tryer | CC BY-SA 4.0 |