Imagine sheep fill a simple (simply connected) polygon $P$, except at one vertex $x$ there is no sheep. One convex vertex $g$ of $P$ is a gate through which the sheep should pass. A herding dog sits at a vertex $x$ and barks. Every sheep point $s$ moves along a radial line from $x$, until $s$ hits the boundary $\partial P$. Then, if the forward vector $v$ from $s$ along $\partial P$ has a strictly positive projection on the vector $xs$, then $s$ moves forward along $v$. If the projection is zero or negative, $s$ is stuck.
The example below shows a polygon $P$ that is not entirely cleared from $x$, because the ray $xa$ is orthogonal to the edge $e$ incident to $g$, and the sheep above that ray are all stuck. (The barking penetrates the fence $\partial P$.) So we could say that $P,g$ is not $1$-herdable from $x$.
However, after the sheep have been cleared from $x$, the dog can walk along the cleared boundary to $x'$, which then clears the remaining sheep because the angle formed by $x' b$ with $e$ is obtuse. So we could say $P,g$ is $2$-herdable from $x$.
I am interested in characterizing the shapes of $P$ that are herdable from some $x$, $k$-herdable for any $k$. For $k=1$, it seems this is necessary and sufficient: $P,g$ is $1$-herdable from $x$ if there is no point $p$ on an edge $e$ such that: (a) $e$ faces $x$, (b) the angle $xp$ makes with $e$ is $\ge 90^\circ$ away from $g$. I realize this formulation is not precise, but perhaps it is clear from the figures.
I would be interested in any description of the shapes of polygons that are $1$- or $2$- or $k$-herdable, and shapes that are definitely not so herdable. For example: All convex polygons with all obtuse angles are $1$-herdable with appropriate choice of $x$ with respect to $g$. But a square is not herdable for any $k$: Some sheep gets stuck in a corner, and the dog cannot occupy the same point as that stuck sheep. (I make an exception for the initial dog vertex $x$.)
Q. Which shapes are, and which shapes are not herdable for any $k$, for any placement of $g$ and $x$?
It is likely too much to hope for a complete characterization, but perhaps large classes of polygon shapes can be settled, for specific $k$.
This notion of herding by repulsion from $x$ is in a sense the inverse of the concept of beacon routing, which relies on attraction. Here is one source among many on that topic:
Shermer, Thomas C. "A combinatorial bound for beacon-based routing in orthogonal polygons." arXiv:1507.03509 (2015).
