Area Problems Involving Kasner Polygons

TL;DR

This paper investigates the area ratios of convex polygons and their Kasner descendants, providing a comprehensive analysis of how the area changes when vertices are placed at specific ratios along edges.

Contribution

It offers a complete characterization of the area ratio between a convex polygon and its Kasner descendant for any fixed ratio m and polygon size n.

Findings

Derived formulas for area ratios of Kasner polygons.

Established bounds for area ratios based on polygon shape.

Generalized results for polygons with any number of sides n.

Abstract

Sequences of polygons generated by performing iterative processes on an initial polygon have been studied extensively. One of the most popular sequences is the one sometimes referred to as {\it Kasner polygons}. Given a polygon $K$, the first Kasner descendant $K'$ of $K$ is obtained by placing the vertices of $K'$ at the midpoints of the edges of $K$. More generally, for any fixed $m$ in $(0,1)$ one may define a sequence of polygons $\{K^{t}\}_{t\ge 0}$ where each polygon $K^{t}$ is obtained by dividing every edge of $K^{t-1}$ into the ratio $m:(1-m)$ in the counterclockwise (or clockwise) direction and taking these division points to be the vertices of $K^{t}$. We are interested in the following problem {\it Let $m$ be a fixed number in $(0,1)$ and let $n\ge 3$ be a fixed integer. Further, let $K$ be a convex $n$-gon and denote by $K'$, the first $m$-Kasner descendant of $K$, that is, the vertices of $K'$ divide the edges of $K$ into the ratio $m:(1-m)$. What can be said about the ratio between the area of $K'$ and the area of $K$, when $K$ varies in the class of convex $n$-gons?} We provide a complete answer to this question.