What is the technical term for an $n$-dimensional generalization of the unit interval $[0, 1]$? Would we call an $n = 1,2,3,...$ dimensional generalization of the unit interval an $n$-cube?
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4Yes. This is a sensible and widely-recognized name. – ncmathsadist Nov 06 '13 at 02:41
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@ncmathsadist Would it also be reasonable to call it an $R^n$ unit interval? The basis of my question is seeing this in print. – user105766 Nov 06 '13 at 02:45
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I'd refer to it as a "cube" or a "cell." – ncmathsadist Nov 06 '13 at 02:47
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I have seen it generalized as an n-dimensional interval, described as the set of points $x_; i=1,2,..,n$ with, e.g., $a_i < x< b_i$, where, of course, you can also have $a_i \leq x_i <b_i$, etc., i.e., closed, semi-open, etc. – user99680 Nov 06 '13 at 02:50
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Most real analysis textbooks will call it an $n$-cube or $n$-rectangle. It comes into play a lot, since the supnorm is equivalent to the Euclidean norm in R and it is sometimes more convenient to do analysis on cubes instead of balls. Integration comes to mind... – Christopher K Nov 06 '13 at 03:06
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Since you said you wanted to generalize the unit interval, you should probably refer to the "unit $n$-cube". You can see that this terminology is used with a google search, which pulls up papers like this one.
Mark S.
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