How do you call a set in which only some of the dimensions are bounded?

Question

Let $\mathbf{X}=(x_1, \dots, x_n)$ be a set in $\mathbb{R}^n$ such that $L_i \leq x_i \leq U_i$ for several $\;i\;$ in $\{1, \dots, n\}$, which means that some of the dimensions of $X$ are bounded (but not all).

Is there a specific name for that property that $X$ meets? Maybe partially bounded?

I know that term semibounded is used to refer to the fact that a given variable has only either an upper or a lower bound, so I think it would not be appropriate to describe the property I described above.

Answer 1

In the context of convex sets I have seen the term "partially bounded" used in a similar way to what you're describing. In particular, let $S$ be a convex set. Then $S$ is partially bounded if there is a finite upper bound on the radii of (hyper)spheres contained in $S$.

Your set $X$ would then be partially bounded in this sense (as we couldn't fit a sphere whose width is larger than the minimum $U_i-L_i$).

Here's a reference that uses it in this way:

http://www.edwardothorp.com/wp-content/uploads/2016/11/PartiallyBoundedSetsOfIinfiniteWidth.pdf