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$π_B$ in asymetric Minkowski normed spaces

Gorka Guardiola Múzquiz

TL;DR

This work extends Golab's classical results on the half-perimeter of unit balls from symmetric normed planes to asymmetric Minkowski spaces with one axis of symmetry. By introducing the offset Minkowski functional $||\cdot||_{B,x_0}$ and carefully defining perimeter notions, the authors establish when $π_B$ is well-defined and compute its values for families of unit balls, notably regular polygons. They show $π_B ≥ 3$ in the general asymmetric setting, with the minimum being attained in several symmetric configurations, and that $π_B$ can take any value in $[3, ∞)$ by appropriate choice of $B$ and centering. The analysis encompasses Radon norms, Beraha-number connections, and offset-center polygons (triangle, square, hexagon), providing both exact formulas and bounds that illuminate the geometry of asymmetric normed planes and their perimeters.

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 $π_B$ for different families of polygons as unit ball B. Then we prove that $π_B \ge 3$, and can take all possible values and is not bounded in such spaces.

$π_B$ in asymetric Minkowski normed spaces

TL;DR

This work extends Golab's classical results on the half-perimeter of unit balls from symmetric normed planes to asymmetric Minkowski spaces with one axis of symmetry. By introducing the offset Minkowski functional and carefully defining perimeter notions, the authors establish when is well-defined and compute its values for families of unit balls, notably regular polygons. They show in the general asymmetric setting, with the minimum being attained in several symmetric configurations, and that can take any value in by appropriate choice of and centering. The analysis encompasses Radon norms, Beraha-number connections, and offset-center polygons (triangle, square, hexagon), providing both exact formulas and bounds that illuminate the geometry of asymmetric normed planes and their perimeters.

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 for different families of polygons as unit ball B. Then we prove that , and can take all possible values and is not bounded in such spaces.

Paper Structure

This paper contains 17 sections, 11 theorems, 73 equations, 17 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Asymmetric Minkowski normed spaces
  3. Radon norms
  4. Regular convex polygons
  5. Simpler expression
  6. Beraha numbers
  7. Polygonal circle numbers
  8. Apothem
  9. Bounds for $\pi_n$
  10. Polygons with an offset center
  11. Offset isosceles triangle
  12. Square with offset center
  13. Regular hexagon with offset center
  14. Minimum $\pi_B$ for the general case
  15. Appendix
  16. ...and 2 more sections

Key Result

Proposition 1.4

If $B$ is a centrally symmetric convex set in $X$ such that each line through $0$ meets $B$ in a non-trivial, closed, bounded segment then $||\cdot||_B$ is a norm in $X$ where $||\cdot||_B$ comes from the next definition, given $B$ meets the properties stated above (proposition 1.1.8 in thompson).

Figures (17)

  • Figure 1: Ball and polygon with an axis of symmetry
  • Figure 2: Triangle as unit ball
  • Figure 3: Constructions to calculate the measure of a side
  • Figure 4: Values of $\pi_{B_n}$ vs. number of sides
  • Figure 5: General construction to measure the side
  • ...and 12 more figures

Theorems & Definitions (32)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Proposition 1.4
  • Definition 1.5
  • Theorem 1.6
  • Definition 1.7
  • Definition 1.8
  • Proposition 1.9
  • Proposition 1.10
  • ...and 22 more