If $\mathbf{K}$ is a compact, convex set with nonempty interior in $d$-dimensional Euclidean space $\mathbb{E}^d$, and $\mathbf{p}$ is an exterior point outside of $\mathbf{K}$, we say that $\mathbf{p}$ 'illuminates' point $\mathbf{q} \in \mathbf{K}$ if the line from $\mathbf{p}$ to $\mathbf{q}$ intersects the interior of $\mathbf{K}$.

Suppose we let $\mathbf{K}_1$ be the set illuminated by $\mathbf{p}$ and $\mathbf{K}_2=\mathbf{K} \setminus \mathbf{K}_1$. My question is: are these two sets ($\mathbf{K}_1$ and $\mathbf{K_2}$) linearly separated by some hyperplane?

If $\mathbf{K}$ is a compact, convex set with nonempty interior in $d$-dimensional Euclidean space $\mathbb{E}^d$, and $\mathbf{p}$ is an exterior point outside of $\mathbf{K}$, we say that $\mathbf{p}$ 'illuminates' point $\mathbf{q} \in \mathbf{K}$ if the line from $\mathbf{p}$ to $\mathbf{q}$ intersects the interior of $\mathbf{K}$.

Suppose we let $\mathbf{K}_1$ be the set illuminated by $\mathbf{p}$ and $\mathbf{K}_2=\delta \mathbf{K} \setminus \mathbf{K}_1$ (where $\delta \mathbf{K}$ is the boundary of $\mathbf{K}$). My question is: are these two sets ($\mathbf{K}_1$ and $\mathbf{K_2}$) linearly separated by some hyperplane?