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To my understanding, a "shape" is a set of points in $n$-dimensional space. e.g., rectangles, triangles, lines, spheres, hyperspheres, etc.. For two (or any amount) of shapes to "intersect", the representative sets for these shapes must have points in common.

Here enters my confusion — if two rectangles are touching on a corner, are they said to be intersecting? Is there a difference between "touching" and "intersecting" in Geometry?

To makes things a little more concrete, consider the following example:

$R1$ and $R2$ are defined by the cross products $\hat{i}\times\hat{j}$ and $\hat{-i}\times\hat{-j}$, respectively. $R1$ and $R2$ thus share only 1 point, the origin $[0,0]$. Are $R1$ and $R2$ said to be intersecting?

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    $\begingroup$ This is not standardized terminology. However, in the context of covering problems, intersect often means intersect on a set of positive measure, which rules out intersecting at a 'corner'. Is your example the one you are actually interested in? If not, it would help to have that extra context, since again, the terminology is not standardized. $\endgroup$ May 21, 2015 at 20:23
  • $\begingroup$ This is interesting, thanks Eric. $\endgroup$
    – Stanley
    May 21, 2015 at 20:25
  • $\begingroup$ @EricStucky Would your definition suggest that the rectangles $\hat{i}\times\hat{j}$ and $\hat{i}\times\hat{-j}$ intersect? Trying to pin down what you mean by a "set of positive measure" $\endgroup$
    – robert
    May 21, 2015 at 20:34
  • $\begingroup$ Adding to what @EricStucky said, consider perpendicular intersecting rectangles like the ones pictured here: mathunion.org/fileadmin/IMU/Logo/pixD.png. Context is important - I would say that these shapes intersect, but the shapes have measure zero in $\mathbb{R}^3$, and even if the rectangles are viewed as positive-measure sets in a plane, the intersection has measure zero. I would expect any mathematical discussion of these ideas that uses a definition of “intersect” other than the usual one for sets ($R1\cup R2\neq\emptyset$) would provide a clear definition. $\endgroup$
    – Steve Kass
    May 21, 2015 at 20:39
  • $\begingroup$ If you're not familiar with the definition, positive measure here means positive area. For example, the area of a shape like a rectangle is a positive number, but a point $(0,0)$ has an area, or measure, of exactly zero. $\endgroup$
    – Stanley
    May 21, 2015 at 21:11

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If sets are closed, they contain their boundaries. In a rectangle, this means that the parallel lines that bound the rectangle are included in the rectangle. So the intersection of $R1$ and $R2$ is non-empty, or in other words, they do intersect.

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  • $\begingroup$ True. We must consider whether we mean open or closed shapes. Good point. $\endgroup$ May 21, 2015 at 20:24
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My understanding has always been that 'intersecting' shapes would, in fact, refer to the set of points. If the two (or more) closed shapes touch on a corner, that particular point is in the sets representing both (or all) shapes and thus we consider them to intersect.

EDIT: Added that touching corners are only sufficient for closed shapes.

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  • $\begingroup$ I added the clarification in the wrong spot. Should be correct now. Thanks. $\endgroup$ May 21, 2015 at 20:26

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