Finite variations on the isoperimetric problem

TL;DR

This paper surveys various isoperimetric problems involving extremal measurements of regions with fixed perimeter, focusing on polygons with limited sides, highlighting known results and open questions.

Contribution

It provides a comprehensive survey of isoperimetric problems for polygons with bounded sides, emphasizing open problems and recent advances.

Findings

Known solutions for convex disks

Open problems for polygons with limited sides

Summary of recent research developments

Abstract

The isoperimetric problem asks for the maximum area of a region of given perimeter. It is natural to consider other measurements of a region, such as the diameter and width, and ask for the extreme value of one when another is fixed. The solution of these problems is known if the competing regions are general convex disks, however several of these problems are still open if the competing regions are polygons with at most a given number of sides. The present work surveys these problems.