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Suppose that $V$ is a (vector) subspace of $\Bbb{R}^{3}$ of dimension $2$ and that $V$ contains an element with all three coordinates non-zero.

To avoid problems with octants sharing axes, suppose the octants to be counted here do not include the axes. I.e., they are of the form $\left\{ (x,y,z): x>0, y>0, z>0 \right\}$ (and similarly for the other seven octants).

What is the maximum number of such octants that $V$ can intersect?

Is there a formula in general for the maximum number of such $n$-dimensional generalized "quadrants" that a $d$-dimensional subspace of $\Bbb{R}^{n}$ can intersect?

References would be especially helpful.

Sum One
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  • By "subspace", do you mean subspace in the usual vector space sense (i.e. contains the origin) or an affine subspace (i.e. can be any translate of a subspace in the vector space sense)? The reason I ask is that the latter seems to be a more geometrically relevant issue one might be interested in. – Dave L. Renfro Feb 26 '23 at 09:23
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    I think I've seen this on the site before; will try to find it later. – joriki Feb 26 '23 at 09:43
  • In $R^3$ for $2$-dim, this is simple right? A plane passing thru zero can intersect at max 4 octants. I am not sure about higher dimensions, or even what a "quadrant" in 3d would mean. – whoisit Feb 26 '23 at 10:09
  • @DaveL.Renfro thank you for your comment. The usual vector space sense. Question clarified to state this. – Sum One Feb 26 '23 at 10:11
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    @joriki maybe this one? https://math.stackexchange.com/questions/1887759/what-is-the-maximum-number-of-quadrants-in-n-dimensional-space-that-a-k-dime?rq=1 Your comment inspired me to look a bit harder. Thanks! – Sum One Feb 26 '23 at 16:01

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One way to think about this is look at the intersections of the coordinate planes with your intersecting plane (i.e. your 2d subspace). Unless your plane is parallel to one of the coordinate axes you get one line of intersection per coordinate plane. Three lines through the origin divide the plane into six sectors. Each of these sectors lies in a different octant.

If your question had been about any plane i.e. affine subspace, and not just about planes through the origin i.e. linear subspace, then you would have gotten three lines that aren't necessarily concurrent but instead form a triangle in the general situation. That triangle would have formed a seventh area of the plane, and the octant opposite the one with the triangle would be the only one you don't intersect.

MvG
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