# Concepts of curvatures in normed planes

## TL;DR

This paper provides a comprehensive overview and new insights into various curvature concepts in normed planes, including characterizations of special curves and introducing a new curvature type, thereby filling a significant gap in the literature.

## Contribution

It systematically reviews existing curvature concepts in normed planes, introduces a new curvature type, and derives new characterizations and properties of curves within this framework.

## Key findings

- Characterizations of curves of constant curvature
- New characterizations of Radon planes and Euclidean subcases
- Analogues of classical theorems like the four vertex theorem

## Abstract

The theory of classical types of curves in normed planes is not strongly developed. In particular, the knowledge on existing concepts of curvatures of planar curves is widespread and not systematized in the literature. Giving a comprehensive overview on geometric properties of and relations between all introduced curvature concepts, we try to fill this gap. Certainly, this yields a basis for further research and also for possible extensions of the whole existing framework. In addition, we derive various new results referring in full broadness to the variety of known curvature types in normed planes. These new results involve characterizations of curves of constant curvature, new characterizations of Radon planes and the Euclidean subcase, and analogues to classical statements like the four vertex theorem and the fundamental theorem on planar curves. We also introduce a new curvature type, for which we verify corresponding properties. As applications of the little theory developed in our expository paper, we study the curvature behaviour of curves of constant width and obtain also new results on notions like evolutes, involutes, and parallel curves.