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Can anyone tell me what $\mathbb R^2_{++}$ means? Is it different from $\mathbb R^2_+$?

Thank you so much!

Edit (answer): This is what the author meant (I found it in the lecture notes by the same professor but of a different course): $\mathbb R^2_{++}=\{(x,y)\in\mathbb R^2\mid x>0, y>0\}$ , while $\mathbb R^2_+=\{(x,y)\in\mathbb R^2\mid x≥0, y≥0\}$ .

Asaf Karagila
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babbo natale
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  • Please confirm that I LaTeX'd your notation properly. – Asaf Karagila Dec 21 '12 at 11:56
  • yes, that's exactly what i meant. thank you – babbo natale Dec 21 '12 at 12:00
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    What do you think $\mathbb{R}^2_+$ means? (It's not entirely obvious.) My guess would be that $\mathbb{R}^2_{++}$ is supposed to denote $(\mathbb{R}_+)^2$, i.e. the set of points $(x,y)$ where both coordinates are positive, but check the text where you found it. There is probably a definition somewhere (at least, there should be). I have never seen this notation before. – mrf Dec 21 '12 at 12:03
  • @mrf: I've used that notation $\mathbb{R}^2_{++}$ previously (and privately only). It is somewhat useful in convex analysis when sometimes one has to consider individual orthants, and so $\mathbb{R}^n_{+++---+\cdots +}$ can be useful sometimes to keep track of the combinatorics. On the other hand, I don't ever use $\mathbb{R}^2_+$: if I want a half-space I call it $\mathbb{R}_+\times \mathbb{R}^n$. – Willie Wong Dec 21 '12 at 12:49
  • As a follow up: in 2D there is a usual convention for numbering the quadrants. (Though I often forget which is second and which is fourth.) But in higher dimensions I am not aware of any established numbering convention... – Willie Wong Dec 21 '12 at 12:50

1 Answers1

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This is not a usual notation, so it should probably be mentioned somewhere, but a reasonable guess would be $\{(x,y)\in\mathbb R^2\mid x>0, y>0\}$.

If you supply more context, and in particular where you have seen this notation perhaps further information can be given.

Asaf Karagila
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  • It comes from an example of utility functions. – babbo natale Dec 21 '12 at 12:14
  • Again, more details on where you found that example will help. (Note that "in a book" or "online" is not a helpful expansion of current knowledge of the whereabouts of this example.) – Asaf Karagila Dec 21 '12 at 12:17
  • It comes from an example of utility functions. Let u:R^2_+ --> R such that u(x)=(a(x_1)^b+(1-a)(x_2)^b)^(1/b) with a in [0,1] anb b in (0,1]. Let v:R^2_++ -->R such that v(x)=(x_1)^c(x_2)^(1-c) with c in [0,1]. This example should show that this 2 utility functions have different domain. – babbo natale Dec 21 '12 at 12:20
  • I found it in some lecture notes, there is no way to ask directly to the professor who wrote it. – babbo natale Dec 21 '12 at 12:23
  • @babbonatale: Asaf's point is that "details about where you found that example" means tell us the author and title of the book, preferrably plus edition and page number. – hmakholm left over Monica Dec 21 '12 at 12:25
  • The point is that there is no book at all. That is why I am so confused. I thought it was a common notation, but I see this is not the case. – babbo natale Dec 21 '12 at 12:28
  • @babbonatale: But where did you find those lecture notes? If you took them from a friend, ask the friend, if you found them online ask whoever posted them to add the information to the notes. First, however, you should look more carefully in the notes. – Asaf Karagila Dec 21 '12 at 12:33
  • Okay I'll do that. At least now I know this is not a widely accepted notation. Thank you very much. – babbo natale Dec 21 '12 at 12:41