Timeline for Integer decomposition of dilated integral polytopes
Current License: CC BY-SA 3.0
6 events
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| May 17, 2015 at 21:05 | comment | added | Francisco Santos | 3. A more constructive proof of the same Lemma is as follows. As before, suppose $P$ is a simplex. Any point $p$ in $dP$ can be written as a (perhaps not integral) combination $\lambda_1 v_1 + \cdots + \lambda_{n+1} v_{n+1}$ of the vertices of $P$, with $\sum_i \lambda_i=d$ and all $\lambda_i\ge 0$. Since $d\ge n+1$, some $\lambda_i$ is $\ge 1$. That means that $p - v_i$ is in $(d-1)P$. Iterate until you get a point at height $\le n$. | |
| May 17, 2015 at 20:57 | comment | added | Francisco Santos | 2. By a change of basis (take $v_1,\dots,v_n$ as your basis) you can think of $C_P$ as the positive orthant and of $Z$ as the unit cube $[0,1]^n$. The positive orthant is tiled by integer-translated copies of the unit cube. | |
| May 17, 2015 at 20:56 | comment | added | Francisco Santos | 1. "pos" means "positive span"; that is, if $A$ is a subset of a real vector space, $pos(A)$ is the cone of linear combinations of elements of $A$ with nonnegative coefficients. | |
| May 17, 2015 at 15:30 | comment | added | hypercube | Thank you for the answer! I'm glad that the result holds! I have a few questions though if you don't mind: 1. What does pos mean in the definition of the cone? 2. Can you explain why the cone is tiled by integer translations of $Z$? 3. How constructive is this process? If I gave you $p$ could you give an algorithm to determine $p'$ and the $p_i$'s? | |
| May 17, 2015 at 15:27 | vote | accept | hypercube | ||
| May 16, 2015 at 19:07 | history | answered | Francisco Santos | CC BY-SA 3.0 |