Summary: ``We study MINLO (mixed-integer nonlinear optimization) formulations of the disjunction $x \in \{0\}\cup[l, u]$, where $z$ is a binary indicator of $x \in [l, u]$ $(u >\ell> 0)$, and $y$ `captures' $f (x)$, which is assumed to be convex on its domain $[l, u]$, but otherwise $y = 0$ when $x = 0$. This model is useful when activities have operating ranges, we pay a fixed cost for carrying out each activity, and costs on the levels of activities are convex. Using volume as a measure to compare convex bodies, we investigate a variety of continuous relaxations of this model, one of which is the convex-hull, achieved via the `perspective reformulation' inequality $y \geq z f (x/z)$. We compare this to various weaker relaxations, studying when they may be considered as viable alternatives. In the important special case when $f (x) \coloneq x^p$, for $p > 1$, relaxations utilizing the inequality $yz^q \geq x^p$, for $q \in [0, p - 1]$, are higher-dimensional power-cone representable, and hence tractable in theory. One well-known concrete application (with $f (x) \coloneq x^2$) is mean-variance optimization (in the style of Markowitz), and we carry out some experiments to illustrate our theory on this application.''