Timeline for Is the direction of the longest line of a polytope unique?
Current License: CC BY-SA 2.5
18 events
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| May 29, 2010 at 1:11 | history | edited | j.c. |
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| Jan 11, 2010 at 14:38 | comment | added | some_random_guy | I edited the question to fix the terminology. | |
| Jan 11, 2010 at 14:35 | history | edited | some_random_guy | CC BY-SA 2.5 |
Edited to fix terminology as per comment by Harald
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| Jan 9, 2010 at 13:49 | comment | added | some_random_guy | The way I have defined u_{max} above - it is a vector. However, all elements of u_{max} have the same value and hence we have a hypercube in p dimensions. X can have positive or negative values. The original context in which the above issue arises would constrain X to have only +1, 0 or -1 values but I did not mention this constraint as I felt that the general problem would have a positive answer. | |
| Jan 9, 2010 at 6:34 | comment | added | 002 | It's not clear to me if u_{max} is a scalar or a vector. The statement could be understood either way. I hope Srikant can clarify. (Also, it would be nice to be sure that X isn't assumed to have positive entries, as often happens in linear programming.) | |
| Jan 8, 2010 at 20:16 | answer | added | j.c. | timeline score: 1 | |
| Jan 8, 2010 at 19:27 | comment | added | j.c. | To expand on Harald's comment a bit, assuming that the rank of X is m, what Srikant is asking is: "What's the longest segment contained inside the polytope formed by intersecting an m-plane (specified by X and y) with the hypercube of side-length u_{max} in R^p? Is the direction of this segment fixed if u_{max} is large enough?" | |
| Jan 8, 2010 at 18:51 | comment | added | Harald Hanche-Olsen | (Argh! I forgot that one can't get LaTeX formatting in comments beyond the fifth.) | |
| Jan 8, 2010 at 18:45 | comment | added | Harald Hanche-Olsen |
Pointwise ordering is the only that makes sense here. I.e., $u\le v$ means $u_i\le v_i$ for each $i$. The terminology is off here, though: Hyperplanes are unbounded (and of dimension one less than the containing space). We are looking at a polytope which is the intersection of an affine subspace with the standard hypercube in $\mathbb{R}^p$.
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| Jan 8, 2010 at 18:37 | comment | added | some_random_guy | @Qiaochu: I am afraid I do not understand your question. My knowledge of math is fairly limited. | |
| Jan 8, 2010 at 18:24 | comment | added | Qiaochu Yuan | What ordering are you using on vectors? Pointwise? | |
| Jan 8, 2010 at 18:20 | comment | added | some_random_guy | @Deane: Cleaned up notation to indicate that the question relates to real spaces. | |
| Jan 8, 2010 at 18:17 | history | edited | some_random_guy | CC BY-SA 2.5 |
added 21 characters in body; added 10 characters in body
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| Jan 8, 2010 at 18:15 | comment | added | some_random_guy | Sorry for the confusion. I do not usually use LaTeX. I cleaned up the LaTeX glitches. | |
| Jan 8, 2010 at 18:14 | comment | added | Deane Yang | Sorry for the dumb question but I work only on vector spaces over the reals: This appears to be a question on a finite discrete space of some sort? | |
| Jan 8, 2010 at 18:12 | comment | added | Deane Yang | Any chance you want to edit the LaTeX symbols? | |
| Jan 8, 2010 at 18:11 | history | edited | some_random_guy | CC BY-SA 2.5 |
deleted 9 characters in body; added 11 characters in body; added 2 characters in body
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| Jan 8, 2010 at 18:06 | history | asked | some_random_guy | CC BY-SA 2.5 |