I am interested in non-convex polygons and extremal properties thereof, specifically on hexagonal (honeycomb) lattices (or three valent and three colorable lattices in general). In particular, I am aware of Reinhardt polygons are those that have the largest perimeter for a diameter, largest width for a diameter, and the largest with for a perimeter. However, for non-convex polygons I cannot find many similar notions and references. Let me fix some terms in the following: with vertices I mean vertices of the (hexagonal) lattice, with edges the edges of the (hexagonal) lattice, and with corners I mean the vertices of the polygon (which are on vertices of the lattice).
The metrics I am interested in are:
- the number of vertices inside the polygon $V$,
- the minimum number of edges between two corners, $A$,
- and the minimum number of edges between two sides of the polygon, $B$.
And the extremal problems I would like to solve are: i) Given a number of vertices, what is the polygon with the maximum number of corners s.t. $A$ and $B$ are maximized, ii) Given a number of corners, what is the polygon with the minimum number of vertices s.t. $A$ and $B$ are maximized, iii) What is the relation between number of corners, number of vertices inside the polygon, and $A$ and $B$ in general. Are there any known methods (open-source implementations) or bounds for these or similar problems? I imagine that standard notions like diameter and width help to find crude bounds, however, in the non-convex case it is unclear to me that this is generally true. Disclaimer: (Computational) Geometry is not my field of education as you might notice from the formulation of this question.
Until now I have thought about considering the triangle as the base case, since it is optimal in the metrics described above. From there I was thinking that more general shapes can be obtained by taking for instance three triangles and placing them on the lattice s.t. they intersect and then taking the polygon as the union of the three triangles. Placing them right allows to get good values for $A$ and $B$, for a number of corners however, I am unsure how this compares to the best possible shape, for instance.