Let $K_1$ and $K_2$ be closed, convex subsets of $\mathbb{R}_{\geq 0}^n$. For $x \in \mathbb{R}_{\geq 0}^n$, let $s(x) = \{i \in \{1,...,n\}: x_i>0\}$ be the support of $x$. I'm trying to better understand the following property:
(*) For all $x \in K_1$ and $y \in K_2$, $s(x) \neq s(y)$.
Obviously, if (*) holds, then $K_1 \cap K_2 = \emptyset$. But the converse isn't true: $K_1$ and $K_2$ can be disjoint and have points with common support.
This is open-ended, but I'm simply looking for a convenient necessary and sufficient condition for (*).
Any hints or references are appreciated.
Referring to my comments above, I think that $(*)$ holds if and only if $K_1$ and $K_2$ are contained in disjoint unions of these maximal classes.
Each such maximal class (in $\mathbb R_{\geq 0}^n$) is the Cartesian product of $n$ factors, each of which is either $\{0\}$ or $\mathbb R_+$. There are precisely $2^n$ such classes. Note that the classes are pairwise disjoint and they exhaust $\mathbb R_{\geq 0}^n$.