# On the polygon determined by the short diagonals of a convex polygon

## TL;DR

This paper proves a conjecture that among convex pentagons, the maximum ratio of the area of the polygon formed by its diagonals to the original is achieved by an affine regular pentagon, and discusses polygons with more vertices.

## Contribution

It confirms the conjecture about the maximum area ratio for convex pentagons and analyzes the case for polygons with six or more vertices.

## Key findings

- Maximum area ratio occurs for affine regular pentagon.
- For polygons with six or more vertices, trivial solutions are optimal.
- The conjecture is rigorously proven.

## Abstract

Let $K$ be a convex pentagon in the plane and let $K_1$ be the pentagon bounded by the diagonals of $K$. It has been conjectured that the maximum of the ratio between the areas of $K_1$ and $K$ is reached when $K$ is an affine regular pentagon. In this paper we prove this conjecture. We also show that for polygons with at least six vertices the trivial answers are the best possible.