Timeline for Taking powers of polytopes
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 18, 2016 at 20:54 | comment | added | gradstudent | How do you define the ``Minkowski product"? | |
| Oct 18, 2016 at 20:40 | comment | added | Steve Huntsman | An only slightly serious observation: taking the tropical structure on $\mathbb{R}^n$ would turn the usual "Minkowski sum" into a different notion of "Minkowski product" than the one I sketched above. Then the square of the unit square is a square with edge length 2. | |
| Oct 18, 2016 at 20:34 | comment | added | Steve Huntsman | For a polytope $P$ in $\mathbb{R}^2$ one could treat $\mathbb{R}^2$ as $\mathbb{C}$ and consider the "Minkowski power" $P^n := \{\prod_{j=1}^n z_j : z_j \in P\}$. But then the square of the unit square is the upper half of a lens shape centered at the origin. | |
| Oct 18, 2016 at 19:28 | answer | added | Liviu Nicolaescu | timeline score: 2 | |
| Oct 18, 2016 at 18:06 | comment | added | Sam Hopkins | Okay, you are right that it does not necessarily make $m^d$ copies, but of course the volume is equal to the volume of $m^d$ copies at least. | |
| Oct 18, 2016 at 18:02 | comment | added | gradstudent | For general polytopes how do you see this happenning? I am having a hard time imagining how a dilation by a factor of $m$ makes $m^d$ copies of a $d$ dimensional polytope. | |
| Oct 18, 2016 at 17:58 | comment | added | Sam Hopkins | I don't think you'll have a good analog of $z \mapsto z^n$ for $d$-dimensional polytope $P$ unless $n = m^d$, in which case dilation by a factor of $m$ does take $n$ copies of $P$ and stitches them together. | |
| Oct 18, 2016 at 17:55 | comment | added | Sam Hopkins | Sorry, I should've said that polytopes are usually studied up to unimodular affine equivalence. | |
| Oct 18, 2016 at 17:54 | comment | added | gradstudent | I don't want to take products. Think of what the map $z^n$ does as a map from the complex plane to itself - it makes $n$ copies of the initial region and stiches them from end to end. One might take a Riemann surface view of this too. Is there an analogue to this in the world of polytopes? | |
| Oct 18, 2016 at 17:53 | comment | added | Sam Hopkins | By the way, the product $P \times Q$ of two polytopes is a well-studied concept: if $P$ is of dimension $p$ and $Q$ is of dimension $q$, then $P\times Q$ is of dimension $p+q$. So there is already a notion of the "power" of a polytope, but this is different than what you want here because it increases the dimension. | |
| Oct 18, 2016 at 17:52 | comment | added | Sam Hopkins | The map that dilates space by a factor of two maps the unit square in the positive orthant to the square with vertices $(0,0),(2,0),(0,2),(2,2)$. This is equivalent to your square via translation, and generally people only study polytopes up to translation (or even, affine equivalence) anyways. Dilation is central to Ehrhart theory, one main topic in the study of polytopes (en.wikipedia.org/wiki/Ehrhart_polynomial). | |
| Oct 18, 2016 at 17:45 | comment | added | gradstudent | Could you explicitly state which map you think does this? Also I used the square as an example. I am looking for something more general. | |
| Oct 18, 2016 at 17:40 | comment | added | Sam Hopkins | Isn't this just "dilation"? | |
| Oct 18, 2016 at 17:27 | history | asked | gradstudent | CC BY-SA 3.0 |