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In a $1$ dimensional object, the name for the region enclosed by it is the length of the object. In a $2$ dimensional object, the name for the region is the area of the object. In a $3$ dimensional object, the name is the volume of the object. What is the name in a $4$ dimensional object? Hypervolume? In general, what can we call the name of this region in a $n$ dimensional object? ("the region enclosed by this $n$ dimensional object" seems too long and wordy).

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  • $\begingroup$ I've seen the term "bulk" used for polychorons, but the general term is indeed "hypervolume". $\endgroup$ – J. M. is a poor mathematician Jun 1 '13 at 6:11
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I would say that just volume is actually a fine choice for any dimension $\geq 3$, but if you really want to emphasize the relevant dimension, you could say $n$-volume or $n$-hypervolume. Take a look at the Wikipedia page on Lebesgue measure.

I think good catch-all terms (i.e., for dimensions $1$ and $2$ as well) would be measure or content.

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  • $\begingroup$ $4$-$volume$ sounds a little strange when said aloud, but I like it. $\endgroup$ – Justin Jun 1 '13 at 6:24
  • $\begingroup$ Glad I could help :) By the way, to get formatting in text, I suggest using Markdown (see here), so that to get $4$-volume, you would write $4$-*volume*, not $4$-$volume$. $\endgroup$ – Zev Chonoles Jun 1 '13 at 6:29
  • $\begingroup$ Thank you, I forgot about that (but I did use it in the question itself). Also, no wonder why I recognized your username; you were one of the old moderators! $\endgroup$ – Justin Jun 1 '13 at 6:30
  • $\begingroup$ Ah, I should have noticed, sorry! $\endgroup$ – Zev Chonoles Jun 1 '13 at 6:31
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How about $n$-volume? Many objects carry the name from 3 dimensions as they are generalized to higher dimensions: $n$-cube, $n$-ball whose boundary is an $(n - 1)$-sphere come to mind.

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