Let $C \subseteq H$ be a cone that contains the origin. This is a set that satisfices $v, w \in C$ then $v + w \in C$ and $\lambda v \in C$ if $\lambda \geq 0$. I am insterested in two types of cones,
"Planar Cones"
$C \subseteq H$ is a cone and $$ C = \{\sum a_iv_i + \sum b_i e_i, \, \quad a_i \geq 0, \, b_i \in \mathbb R\}$$ where $v_1, e_1, v_2, e_2, \ldots$ are lineraly independent.
Finitely Planar Cones
C is a cone and $D = C \cap S$ is a Planar Cone for any $S \subseteq H$ finite dimentional subspace.
Does the set of non decreasing functions over an interval hold any of these properties?