Timeline for Given a finite set of points, does there exist a linear function pass through a point and strictly below the other points for all the points?
Current License: CC BY-SA 4.0
8 events
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| Apr 20, 2020 at 20:39 | comment | added | Francis | I think another problem here is that my $x$ is in $d$ dimension. Therefore, we don't have such kind of "neighboring relationship from Nik's comment" for the points, i.e. we don't have an ordering for the $\{x_t\}_{t\in T}$ so we don't have neighboring points for each point. | |
| Apr 15, 2020 at 3:22 | comment | added | Francis | To be more specific, my question is like the following: given variable sequence $\{ x_t \}_{t \in [T]} $ and function value sequence $\{ y_t \}_{t \in [T]} $, when would it be possible to find a concave function $u$ such that $u(x_t) = y_t$. | |
| Apr 15, 2020 at 2:37 | comment | added | Francis | Nik's understanding is correct, but my question is what kind of sequences of $X$ and $Y$ can make this condition (i.e. each point should lie below the line joining the two points to either side of it) be satisfied? For example, give some points $\{(0,1), (1,0), (2,1)\}$ (i.e. $f(x) = (x-1)^2$), we know we can construct 3 linear functions as desired. However, if we are given a set of $\{(0,-1), (1, 0), (2, -1)\}$ then we can't construct such linear functions. Therefore, my question is: if we are given a set of points, can we construct such linear functions? | |
| Apr 14, 2020 at 22:19 | comment | added | LSpice | MathJax supports links, so I combined the separate text-and-URLs into a single link. Since you were concerned with pointing to a specific revision, I linked that, too. Also, the problem has two answers; I picked the one that seemed more likely and linked that. I hope that all this is agreeable. | |
| Apr 14, 2020 at 22:18 | history | edited | LSpice | CC BY-SA 4.0 |
Inlined lots of URLs
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| Apr 14, 2020 at 21:09 | comment | added | Nik Weaver | Maybe I don't understand the question ... isn't the condition just that each point should lie below the line joining the two points to either side of it? | |
| Apr 14, 2020 at 20:23 | history | edited | Francis | CC BY-SA 4.0 |
added 8 characters in body
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| Apr 14, 2020 at 20:16 | history | asked | Francis | CC BY-SA 4.0 |