  We study MINLO (mixed-integer nonlinear optimization) formulations of the
disjunction $x\in\{0\}\cup[\ell,u]$, where $z$ is a binary indicator of
$x\in[\ell,u]$ ($0 \leq \ell <u$), and $y$ "captures" $f(x)$, which is assumed
to be convex and positive on its domain $[\ell,u]$, but otherwise $y=0$ when
$x=0$. This model is very useful in nonlinear combinatorial optimization, where
there is a fixed cost of operating an activity at level $x$ in the operating
range $[\ell,u]$, and then there is a further (convex) variable cost $f(x)$. In
particular, we study relaxations related to the perspective transformation of a
natural piecewise-linear under-estimator of $f$, obtained by choosing
linearization points for $f$. Using 3-d volume (in $(x,y,z)$) as a measure of
the tightness of a convex relaxation, we investigate relaxation quality as a
function of $f$, $\ell$, $u$, and the linearization points chosen. We make a
detailed investigation for convex power functions $f(x):=x^p$, $p>1$.
