  In decision-making problems under uncertainty, probabilistic constraints are
a valuable tool to express safety of decisions. They result from taking the
probability measure of a given set of random inequalities depending on the
decision vector. Even if the original set of inequalities is convex, this
favourable property is not immediately transferred to the probabilistically
constrained feasible set and may in particular depend on the chosen safety
level. In this paper, we provide results guaranteeing the convexity of feasible
sets to probabilistic constraints when the safety level is greater than a
computable threshold. Our results extend all the existing ones and also cover
the case where decision vectors belong to Banach spaces. The key idea in our
approach is to reveal the level of underlying convexity in the nominal problem
data (e.g., concavity of the probability function) by auxiliary transforming
functions. We provide several examples illustrating our theoretical
developments.
