Extremal Values of Pi

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 $\pi$.