Affine Properties of Convex Equal-Area Polygons

TL;DR

This paper explores affine geometric properties of convex equal-area polygons, establishing discrete analogs of classical affine differential geometry concepts and inequalities, and demonstrating their approximation capabilities for smooth convex curves.

Contribution

It introduces discrete affine differential geometry concepts for convex equal-area polygons, including affine normal, curvature, and analogs of classical theorems and inequalities.

Findings

Discrete affine normal and curvature definitions for polygons

Discrete six vertices theorem and affine isoperimetric inequality

Approximation of smooth convex curves by convex equal-area polygons

Abstract

In this paper we discuss some affine properties of convex equal-area polygons, which are convex polygons such that all triangles formed by three consecutive vertices have the same area. Besides being able to approximate closed convex smooth curves almost uniformly with respect to affine length, convex equal-area polygons admit natural definitions of the usual affine differential geometry concepts, like affine normal and affine curvature. These definitions lead to discrete analogous of the six vertices theorem and an affine isoperimetric inequality. One can also define discrete counterparts of the affine evolute, parallels and the affine distance symmetry set preserving many of the properties valid for smooth curves.