  Recently the authors have explored new concepts of plurisubharmonicity and
pseudoconvexity, with much of the attendant analysis, in the context of
calibrated manifolds. Here a much broader extension is made. This development
covers a wide variety of geometric situations, including, for example,
Lagrangian plurisubhamonicity and convexity. It also applies in a number of
non-geometric situations. Results include: fundamental properties of
$P^+$-plurisubharmonic functions, plurisubharmonic distributions and
regularity, $P^+$-convex domains and $P^+$-convex boundaries, topological
restrictions on and construction of such domains, continuity of upper
envelopes, and solutions of the Dirichlet problem for related Monge-Ampere-type
equations.
  Many results in this paper have been generalized in recent work of the
authors. However, this article covers many cases of geometric interest, and
certain convexity assumptions here allow the use of classical analytic methods,
making the exposition more accessible.
