# Area and perimeter foliations on spaces of polygons

## TL;DR

This paper characterizes families of star-shaped polygons with fixed area and perimeter as leaves of a foliation in the space of polygons, and explores geometric properties of convex polygons related to these measures.

## Contribution

It provides a complete description of polygon families with given area and perimeter, linking geometric properties to these measures.

## Key findings

- Families of star-shaped polygons form a foliation with prescribed area and perimeter.
- Convex polygons' inscriptibility and regularity relate to their perimeter and area.
- The geometric properties of polygons are analyzed in the context of the foliation structure.

## Abstract

We describe all families of star-shaped n-polygons in the Euclidean plane with prescribed perimeter and area ; they are leaves of a foliation F on the space of star-shaped n-polygons. By the way, we study some geometric properties of convex polygons, for instance their inscriptibility in a circle and their regularity in relation with the perimeter and the area.