Extremal Values of Pi

Abstract

We give an introduction to the classical results of Stanisław Gołąb, on the values that pi may attain in arbitrary normed planes, including a classification of the extremal values. This fascinating topic is accessible to undergraduates, but introductory accounts tend to elide the fussy details of classifying the extremal values; we instead give the necessary background to understand the full proofs, with limited outside references. We then reprove a result due to J. Duncan, D. Luecking, and C. McGregor, which states that any norm with quarter-turn symmetry has pi-value at least We also classify when this lower bound is obtained. Finally, we define Radon norms and note an analogy that we hope will help motivate further results.