Necessary and sufficient
conditions for a linearly compact simplex K to be uniquely decomposable at a face
are given. If P is a cone having the Riesz decomposition property and if −f,g are
subadditive homogeneous functions on P with f ≧ g then it is shown that there is an
additive homogeneous function h on P with f ≧ h ≧ g. If P is a lattice cone for the
dual space of an ordered Banach space X and if −f,g are also w∗-continuous then,
under certain conditions, it is possible to choose h ∈ X; a consequence of
this result is Andô’s theorem, that an ordered Banach space has the Riesz
decomposition property if its dual space is a lattice. A nonmeasure theoretic proof of
Edwards’ separation theorem for compact simplexes is also deduced from these
results.
