  MINLO (mixed-integer nonlinear optimization) formulations of the disjunction
between the origin and a polytope via a binary indicator variable have broad
applicability in nonlinear combinatorial optimization, for modeling a fixed
cost $c$ associated with carrying out a set of $d$ activities and a convex
variable cost function $f$ associated with the levels of the activities. The
perspective relaxation is often used to solve such models to optimality in a
branch-and-bound context, especially in the context in which $f$ is univariate
(e.g., in Markowitz-style portfolio optimization). But such a relaxation
typically requires conic solvers and are typically not compatible with
general-purpose NLP software which can accommodate additional classes of
constraints. This motivates the study of weaker relaxations to investigate when
simpler relaxations may be adequate. Comparing the volume (i.e., Lebesgue
measure) of the relaxations as means of comparing them, we lift some of the
results related to univariate functions $f$ to the multivariate case. Along the
way, we survey, connect and extend relevant results on integration over a
simplex, some of which we concretely employ, and others of which can be used
for further exploration on our main subject.
