How does this generally work in high dimensions? In 1d, an interval is a really simple thing. In higher dimensions, it generalizes in a bunch of ways, from a bo

How does this generally work in high dimensions? In 1d, an interval is a really simple thing. In higher dimensions, it generalizes in a bunch of ways, from a box to an arbitrary convex region. Given that practical problems tend to have the interesting bits constrained around a lower dimensional region, I imagine that the wrong representations (e.g. just an axis-aligned box) could make the intervals meaningless. Is that the case? Is it simple to overcome that (e.g. just align the box along the interesting subspace), or in practice is it harder to have useful high dimensional "intervals?"