Timeline for Is there always a maximum anti-rectangle with a corner square?
Current License: CC BY-SA 3.0
30 events
| when toggle format | what | by | license | comment | |
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| Nov 17, 2013 at 17:15 | history | bounty ended | Erel Segal-Halevi | ||
| Nov 15, 2013 at 21:00 | comment | added | mhum | @domotorp I think I sort of see what's going on now. Part of it was the looseness of the description was a little confusing to me. I just put up an answer that tries to formalize the correspondence a little better. | |
| Nov 15, 2013 at 20:00 | comment | added | domotorp | You wrote that "mhum pointed out that one may have several pairs of R-corners opposing a single pair P of L-corners" - I do not think this is possible, at least if a pair of R-corners is matched to a pair of L-corners, then the LL side's "shadow" has to fall on the RR side. So your proof is complete. It also gives 4 good RR sides, which implies 2 corners (unless C is a rectangle). | |
| Nov 15, 2013 at 16:49 | comment | added | domotorp | I believe that this proof can be made rigorous but I feel like one more observation is missing. Also, I think it proves the stronger statement that unless the polygon is a rectangle, there is a maximum antirectangle that has at least TWO corner squares. | |
| Nov 15, 2013 at 13:47 | comment | added | Nick Gill | let us continue this discussion in chat | |
| Nov 15, 2013 at 13:39 | comment | added | Erel Segal-Halevi | @NickGill Please see the answer I added just now, which attempts to complement/simplify your proof. Do you think it is correct? If it is, then you can merge it with your answer. | |
| Nov 15, 2013 at 12:48 | comment | added | Erel Segal-Halevi | Using the minimal RR side is probably not a good idea. In the following shape: i.sstatic.net/vYUhL.png the minimal RR side (at the west) does not have a corner contained in exactly one maximal rectangle, but the larger RR sides (at the east) do. | |
| Nov 15, 2013 at 9:56 | comment | added | Nick Gill | @Erel, there might be an even shorter LL side that is opposite this RR side - so neither of the R corners would be in a unique maximal subrectangle. (This `shortest side' idea was the approach that I initially thought of, but I couldn't make it work and ended up doing the count that I wrote down. Something like this might work of course, but I couldn't figure it out.) | |
| Nov 15, 2013 at 9:54 | comment | added | Erel Segal-Halevi | @NickGill and if we take the shortest RR side? | |
| Nov 15, 2013 at 9:54 | history | edited | Nick Gill | CC BY-SA 3.0 |
response to comment.
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| Nov 15, 2013 at 9:51 | comment | added | Nick Gill | @mhum: your comment about more than one pair of RR-corners being matched with a pair of LL-corners is a valid point - but I think such a situation would result in their being LL-corners in between the RR-corners and our count would not be affected. Thoughts? | |
| Nov 15, 2013 at 9:51 | comment | added | Nick Gill | @domotorp, I think your criterion does guarantee that the middle R-corner has the property we want. I still feel hopeful, though, that we can just work with the criterion for 2 R-corners. | |
| Nov 15, 2013 at 9:47 | history | edited | Nick Gill | CC BY-SA 3.0 |
pointed out error
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| Nov 15, 2013 at 9:46 | comment | added | Nick Gill | @Erel, I don't think you're criterion quite works because we need the shortest side to lie between RR-corners. LL-corners won't work. | |
| Nov 15, 2013 at 9:42 | comment | added | Erel Segal-Halevi | @mhum Good example, thanks. What do you think about my criterion - the shortest RR/LL side - is it always true that at least one corner in this side is contained in exactly one maximal rectangle? | |
| Nov 15, 2013 at 7:23 | comment | added | domotorp | I was also thinking along these lines and believe that this approach has a chance to work. As a first correction, as pointed out my mhum, we need something stronger. How about that the 3 R-corners can be covered by one rectangle? | |
| Nov 15, 2013 at 3:17 | comment | added | mhum | Also, in your proof, you're matching pairs of R-corners to pairs of L-corners. But, how do you ensure that each of the matched L-corners is only matched once? I.e.: how do you know a different pair of R-corners isn't also matched up against one (or both) of these previously matched L-corners? | |
| Nov 14, 2013 at 19:00 | comment | added | mhum | I may not be understanding correctly, but is it necessarily the case that 3 R-corners guarantees the property you're looking for? In this figure i.imgur.com/vI128yH.png, I think the red corners form 3 consecutive R-corners and none of them seem to be contained in a unique maximal rectangle. | |
| Nov 14, 2013 at 18:13 | comment | added | Erel Segal-Halevi | Consider all sides with either RR or LL corners, and select a shortest one. There can be no crenellations on the opposite side. Do you think this is sufficient? | |
| Nov 14, 2013 at 13:21 | comment | added | Nick Gill | exactly right... | |
| S Nov 14, 2013 at 13:09 | history | suggested | Erel Segal-Halevi | CC BY-SA 3.0 |
change "unique" to "exactl
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| Nov 14, 2013 at 12:55 | review | Suggested edits | |||
| S Nov 14, 2013 at 13:09 | |||||
| Nov 14, 2013 at 12:51 | comment | added | Erel Segal-Halevi | Ah, sorry, I probably misunderstood the word "unique" (which means that d lies only in a single maximal subrectangle). | |
| Nov 14, 2013 at 12:43 | comment | added | Nick Gill | in the example you've shown your square $d$ doesn't lie in a unique maximal subrectangle... | |
| Nov 14, 2013 at 11:48 | comment | added | Erel Segal-Halevi | I am stuck at your preliminary claim: "We can replace this square by d and still have an anti-rectangle". See the C-shape image I just added to the question: i.sstatic.net/17bKQ.png There is a unique maximal subrectangle C (dashed), and there is a little square d that lies in it, and there is an anti-rectangle that contains another square in C (3 blue squares), but if we replace this square with d, the result is not an anti-rectangle anymore. | |
| Nov 14, 2013 at 9:05 | history | edited | Nick Gill | CC BY-SA 3.0 |
typo
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| Nov 14, 2013 at 8:56 | history | edited | Nick Gill | CC BY-SA 3.0 |
added a proof
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| Nov 13, 2013 at 20:51 | history | undeleted | Nick Gill | ||
| Nov 13, 2013 at 20:47 | history | deleted | Nick Gill | via Vote | |
| Nov 13, 2013 at 20:33 | history | answered | Nick Gill | CC BY-SA 3.0 |
