Vertex Classification of Planar C-polygons

TL;DR

This paper investigates the boundary singularities of $C$-polygons formed by intersections of homothets of a convex domain, establishing bounds based on the number of homothets and boundary singularities.

Contribution

It provides new bounds on the number of boundary singular points of $C$-polygons and translative $C$-polygons depending on the number of homothets and boundary singularities.

Findings

Number of singular boundary points of a $C$-polygon is between $n$ and $2(n-1)+m$.

For translative $C$-polygons, the singular points count is between $n$ and $n+m$.

Bounds depend on the number of homothets and boundary singular points of $C$.

Abstract

Given a convex domain $C$, a $C$-polygon is an intersection of $n\geq 2$ homothets of $C$. If the homothets are translates of $C$ then we call the intersection a translative $C$-polygon. This paper proves that if $C$ is a strictly convex domain with $m$ singular boundary points, then the number of singular boundary points a $C$-polygon has is between $n$ and $2(n-1)+m$. For a translative $C$-polygon we show the number of singular boundary points is between $n$ and $n+m$.