# Determining whether a path must pass through a given set

I have sets $E^1, \dots, E^n \subset \mathbb{R}^n_{\ge 0}$, none of which contain $0$. I would like to determine whether or not these sets form an "overhang" of $0$. Intuitively speaking, this is an impassible boundary that stops us from drawing a path within $(E^1 \cup \dots \cup E^n)^C$ from $0$ to an arbitrarily high point. Mathematically speaking, I would like to know whether or not there exists a simply-connected set $C \subset E^1 \cup \dots \cup E^n$ such that for any vector $x \ge 0$, there exists a scalar $\lambda > 0$ such that $\lambda x \in C$.

What sort of tests can I run to see if this is the case?

If it helps, we can assume that every $E^j$ is connected, convex, and open.

For my purposes, inequalities on $\mathbb{R}^n$ are pointwise: $a \ge b$ iff $a_i \ge b_i$ for all $i$.

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