Nope. Roughly speaking, pareto-optimality tells us about a gradient, while concavity/convexity tell us about curvature. That said, if we can randomize decisions, then pareto-optimality under expectations implies concavity: if our frontier is convex, we can take a random mix of two points on the frontier in order to get a point whose expected value is a pareto improvement over the frontier.
doesn’t pareto-optimal imply lack of convexity/concavity?
Nope. Roughly speaking, pareto-optimality tells us about a gradient, while concavity/convexity tell us about curvature. That said, if we can randomize decisions, then pareto-optimality under expectations implies concavity: if our frontier is convex, we can take a random mix of two points on the frontier in order to get a point whose expected value is a pareto improvement over the frontier.