Involutes of Polygons of Constant Width in Minkowski Planes

TL;DR

This paper extends concepts of involutes and polygons of constant width to Minkowski planes, demonstrating convergence of involute sequences to symmetric polygons with a shared center.

Contribution

It introduces Minkowskian curvature, evolutes, and involutes for polygons of constant width, and proves convergence properties analogous to the smooth case.

Findings

Sequences of involutes converge to symmetric polygons.

Properties of smooth constant width polygons are preserved in the polygonal case.

The involute iteration process identifies a central point of the polygon.

Abstract

Consider a convex polygon P in the plane, and denote by U a homothetical copy of the vector sum of P and (-P). Then the polygon U, as unit ball, induces a norm such that, with respect to this norm, P has constant Minkowskian width. We define notions like Minkowskian curvature, evolutes and involutes for polygons of constant U-width, and we prove that many properties of the smooth case, which is already completely studied, are preserved. The iteration of involutes generates a pair of sequences of polygons of constant width with respect to the Minkowski norm and its dual norm, respectively. We prove that these sequences are converging to symmetric polygons with the same center, which can be regarded as a central point of the polygon P.