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 $\pi .$ 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.

MSC::

• Primary 52A40
• Secondary 52A10
• 52A38

Acknowledgment

I would like to extend my thanks to Cornelia Van Cott for the talk that inspired this article and the encouragement that helped it come to fruition. I would also like to thank the referees for their kind comments and helpful references.

Notes

1 We write $\text{conv}(A)$ to denote the convex hull of A, which here is simply the polygon with these vertices.

2 In practice, all of the paths that we consider will be continuous, but we don’t require this in the definition.

3 A typical example is the Koch snowflake. In general, a continuous, injective path or loop has finite length if and only if its Hausdorff dimension is 1 (or 0, in the case when the “path” is just a single point) [4].

4 The original paper [7] is only available as a physical copy in a few libraries, making it fairly inaccessible, particularly during a pandemic. As such, most authors reference [7] by way of a French-language summary, which has had the side-effect of dissociating the paper (and even the journal name) from its original language. While [11] does not credit Gołąb with these results, it reproduces all the proofs (making no claim to originality) and sets forth fascinating new avenues of study by generalizing ϖ to higher dimensions in multiple ways.

5 For any vectors $p,q\in {ℝ}^2$, we have p + q = u if and only if u is the midpoint of $[2p,2q]$.

6 Although it bears Birkhoff’s name, this concept goes back at least to Carathéodory [2].

7 When n = 2, Lemma 3.7(b) shows that this notion of “tangency” is precisely that of a supporting line.

8 A more coordinate-free definition is $X\in \mathcal{Q}$ if and only if SX = X for some $S\in \text{GL}(2,ℝ)$ of order 4.

9 The inequality in Theorem 4.4(a) is also proved in [6, Theorem 5], in part, using an inequality from [5]. While it is not mentioned in [6], the main result of [5] also includes an equality case, which can be combined with [6, Theorem 5] to yield an alternative proof of the equality case in Theorem 4.4(a).

10 The accessible survey [2] gives a lovely account of some aspects of the theory that we have totally ignored, and contains the original references for this result, which is quoted therein as Corollary 4. This result is also discussed in the more comprehensive survey [10], where it is Proposition 48.

Additional information

Notes on contributors

Nikhil Henry Bukowski Sahoo

Nikhil Henry Bukowski Sahoo hails from the East Bay, where he attended the Peralta Community Colleges and where he eventually earned a B.A. in mathematics from the University of California, Berkeley. He is currently a Ph.D. student in mathematics at Cornell University studying equivariant symplectic topology.