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Perhaps this isn't the best place to ask, but I'm trying to figure out the proper name for the region that exists in-between two points in higher dimensions. Under certain conditions, n-cube or hypercube would be appropriate, but I'm looking for a more general name.

For example, if the point coordinates are (1,1) and (-1,-1), then the points define a square. The points (-1,1) and (1,-1) define the same region. However, if the point coordinates are (1,1,1) and (-1,-1,-1), then the points define a cube. However, if the points were (1,1,1) and (-1,-1,0), then region defined isn't quite a cube. Same idea applies with more than 3 dimensions.

I ask because I'm trying to name such a space, preferably with 1-2 words instead of a 3+ word phrase.

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"N-dimensional box" or "box" for short. –  Sergei Dec 5 '12 at 0:42

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Your are looking for "$n$-dimensional interval" (each coordinate is in a one-dimensional interval).

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It is called Hyperrectangle or Box

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