$\pi_B$ in asymetric Minkowski normed spaces

TL;DR

This paper extends classical geometric results about the value of pi in normed planes to asymmetric norms with one axis of symmetry, characterizing possible pi values and establishing bounds.

Contribution

It generalizes the concept of pi in asymmetric Minkowski spaces, providing characterizations and bounds for pi_B in these settings.

Findings

pi_B can take all values greater than or equal to 3

pi_B is unbounded in asymmetric Minkowski spaces

Values of pi_B depend on the shape of the unit ball B

Abstract

We extend the classical results of Stanislaw Golab, on the values of pi in arbitrary normed planes, to asymmetric norms where the unit ball has one axis of symmetry. First, we characterize the values of $\pi_B$ for different families of polygons as unit ball B. Then we prove that $\pi_B \ge 3$, and can take all possible values and is not bounded in such spaces.