Content:
The (feasible) set determined by a chance constraint isn't necessarily convex. In his pioneering work in the early 70's, Prekopa has provided sufficient conditions that involve a requirement of quasi-concavity for the probability measure. This lead to the investigation, by Prekopa, Leindler, Borell and others, of the conditions under which a density function generates a quasi-concave measure. Sections 2 and 3 of this paper are devoted to a review of this work. In particular, it provides a much streamlined version of the proof that a logconcave density generates a logconcave measure. The proof also shows that `longconcavity' actually holds for a much larger class of sets than just the class of nonempty measurable convex sets.
The last section of the paper deals with a justification of sampling when dealing with a chance constraint. The law of large numbers for random lsc functions is exploited to obtain the almost sure convergence of the optimal solutions.
June 5, 1997