Timeline for About hyperplanes cutting the discrete hypercube
Current License: CC BY-SA 3.0
10 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Feb 14, 2017 at 2:37 | comment | added | gradstudent | Lets say it this way : Is there any inequality version of the Sperner's theorem? Like saying "At most so many solutions can be found on the $\{-1,1\}^n$ of the inequation $\vec{a}.\vec{x} < b$". Then say I am trying to count solutions to such inequations that live on the $p$ dimensional faces. | |
| Feb 14, 2017 at 1:16 | comment | added | Pat Devlin | That still doesn't make sense to talk about "probability." As for extremal bounds, some hyperplanes (e.g., $x_1 =1$) contain faces entirely, and some (e.g., $x_1 =0$) cut things in half very neatly. What more specifically are you really trying to ask? (Perhaps it would help us understand what you want if you describe your motivation.) | |
| Feb 13, 2017 at 22:16 | history | edited | gradstudent | CC BY-SA 3.0 |
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| Feb 13, 2017 at 22:10 | comment | added | gradstudent | Maybe this is a way to think of what I am trying to convey : Fix the choice of $\vec{a}$ and $\vec{b}$. Now can one show that there is an universal upper bound on the probabilities I am looking for? (...for example see what happens in say the Sperner's Theorem..there is an universal upperbound on the number of solutions of a linear equation on the discrete hypercube and that upperbound doesn't depend on the $\vec{a}$ or $b$ that one starts with..) | |
| Feb 13, 2017 at 20:53 | comment | added | Robert Israel | In that case, there is no uniform random choice. You would need to specify the distribution. And if that distribution allows $b$ to be large compared to $|\vec{a}|$, it is very likely that the hyperplane will miss your hypercube entirely. | |
| Feb 13, 2017 at 20:22 | comment | added | gradstudent | By "hyperplane" I mean any affine space. Consider any function of the form, $\vec{a}.\vec{x}+b =0$. | |
| Feb 13, 2017 at 19:33 | comment | added | Robert Israel | Do you mean a hyperplane through the origin? Otherwise, there's no such thing as a random hyperplane. | |
| Feb 13, 2017 at 16:40 | history | edited | gradstudent | CC BY-SA 3.0 |
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| Feb 13, 2017 at 16:34 | history | asked | gradstudent | CC BY-SA 3.0 |