Convexity via weak distributive laws

We study the canonical weak distributive law d of the powerset monad over the semimodule monad for a certain class of semirings containing, in particular, positive semifields. For this subclass we characterise d as a convex closure in the free semimodule of a set. Using the abstract theory of weak distributive laws, we compose the powerset and the semimodule monads via d, obtaining the monad of convex subsets of the free semimodule.