Extremal Values of Pi

TL;DR

This paper explores the extremal values of pi in normed planes, revisiting classical results, and characterizes when a normed plane is Euclidean based on symmetry and pi-value.

Contribution

It provides a new classification of extremal pi-values in normed planes and characterizes Euclidean norms via quarter-turn symmetry and pi-value.

Findings

Norms with quarter-turn symmetry have pi-value ≥ π.

A norm is Euclidean iff it has quarter-turn symmetry and pi-value = π.

Reproves and extends classical results on pi in normed planes.

Abstract

We discuss the classical results of Stanis{\l}aw Go\l\k{a}b, on the values of pi in arbitrary normed planes, including the classification of extremal values. We reprove the result of J. Duncan, D. Luecking, and C. McGregor, which states that any norm with quarter-turn symmetry has pi-value $\geq \pi$. We also show that a norm is Euclidean iff it has quarter-turn symmetry in some basis and has pi-value $\pi$.