The morphology of convex polygons (Q1205915)

A vertex of a simple polygon \(P\) is said to be unimodal if the Euclidean distance function to the other vertices of \(P\) is unimodal. The author defines three very special classes of convex polygons with the following hierarchy: Semi-circle \(\subset\) weakly semicircle \(\subset\) barn-shaped. All vertices of a semi-circle polygon are unimodal. Weakly semi-circle polygons must have two unimodal vertices but not more than two, and barn- shaped polygons must have one unimodal vertex but not more. All proves are very elementary.