  Idempotent analogues of convexity are introduced. It is proved that the
category of algebras for the capacity monad in the category of compacta is
isomorphic to the category of $(\max,\min)$-idempotent biconvex compacta and
their biaffine maps. It is also shown that the category of algebras for the
monad of sup-measures ($(\max,\min)$-idempotent measures) is isomorphic to the
category of $(\max,\min)$-idempotent convex compacta and their affine maps.
