In the previous question I asked:
Given a polygon with n vertices, how small can n be before it is impossible to have an interior point p such that the distance from any vertex b to p is greater than the distance from b to its nearest adjacent vertex, b'?
For example, for any triangle, it seems that an interior point will be closer to at least one vertex than that vertex is to its nearest neighbor.
From my own inspection, it seems like the answer might be 6 - a regular hexagon is the first shape for which a point at the center is the same distance from each vertex as the vertices are from their nearest neighbors.
However, I can't figure out how to formulate a rule, or apply the intuition from the regular hexagon to irregular shapes.
For my use case, it would be useful to know "below n vertices, there is no way to have such an interior point."
My updated question is: how small can $n$ be before it is impossible to have a point $p$ such that the distance from any vertex $b$ to $p$ is greater than the distance from $b$ to either of its adjacent vertices, $b'$?