  MINLO (mixed-integer nonlinear optimization) formulations of the disjunction
between the origin and a polytope via a binary indicator variable is broadly
used in nonlinear combinatorial optimization for modeling a fixed cost
associated with carrying out a group of activities and a convex cost function
associated with the levels of the activities. The perspective relaxation of
such models is often used to solve to global optimality in a branch-and-bound
context, but it typically requires suitable conic solvers and is not compatible
with general-purpose NLP software in the presence of other classes of
constraints. This motivates the investigation of when simpler but weaker
relaxations may be adequate. Comparing the volume (i.e., Lebesgue measure) of
the relaxations as a measure of tightness, we lift some of the results related
to the simplex case to the box case. In order to compare the volumes of
different relaxations in the box case, it is necessary to find an appropriate
concave upper bound that preserves the convexity and is minimal, which is more
difficult than in the simplex case. To address the challenge beyond the simplex
case, the triangulation approach is used.
