Timeline for Number of Hyper-cube cuts
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Mar 2, 2012 at 21:00 | comment | added | Gerhard Paseman | While I am here, I may as well make the last part explicit. A growth rate polynomial in the number of vertices would imply a bounded multiplicative factor in the (growth of the) number of cuts as the dimension went up by 1. However n/2 is not a bounded multiplicative factor. Gerhard "Ask Me About System Design" Paseman, 2012.03.02 | |
| Mar 2, 2012 at 20:53 | comment | added | Gerhard Paseman | Further, you will find that the procession of cuts is not arbitrary; each such must at some point duplicate the cut of the lower cube. This restriction leads me to guess that Aaron has the right approximation, but I do not have a proof. Gerhard. "Ask Me About System Design" Paseman, 2012.03.02 | |
| Mar 2, 2012 at 20:49 | comment | added | Gerhard Paseman | Some unhandwaving: For a given cut of the lower cube, find a hyperplane that achieves it. Wiggle it slightly so that it is not parallel to any line joining two points of that hypercube. After its relationship with the lower cube is established, you can still rotate it so that, as it moves, it intersects exact one point of the upper cube in turn. This gives at least 2^d distinct cuts. Gerhard "Ask Me About System. Design" Paseman, 2012.03.02 | |
| Mar 2, 2012 at 19:58 | comment | added | John Jiang | Why can you pair with $2^d$ cuts from the parallel d-cube? Are these labelled by the vertices of the parallel cube? It seems to me some of them might overlap. | |
| Feb 29, 2012 at 23:24 | vote | accept | Robert | ||
| Feb 29, 2012 at 22:34 | comment | added | Gerhard Paseman | Also note that any cut of the copy must preserve come of the original: both pieces in the copy cannot both intersect the sets induced from the original cut. This suggests to me that Aaron has the right order of magnitude in terms of the number of vertices. Gerhard "Ask Me About Cutting Remarks" Paseman, 2012.02.29 | |
| Feb 29, 2012 at 19:18 | history | answered | Gerhard Paseman | CC BY-SA 3.0 |