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On the Banach-Mazur Ellipse

V. D. Babev, M. Ivanov, R. Nikolov

Abstract

We provide a new proof of Ader's characterisation of the ellipse of minimal Banach-Mazur distance to the unit circle of a normed plane in terms of contact and extremal points. Our method reveals the relation of this problem to the Chebyshev alternance.

On the Banach-Mazur Ellipse

Abstract

We provide a new proof of Ader's characterisation of the ellipse of minimal Banach-Mazur distance to the unit circle of a normed plane in terms of contact and extremal points. Our method reveals the relation of this problem to the Chebyshev alternance.
Paper Structure (3 sections, 10 theorems, 71 equations)

This paper contains 3 sections, 10 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. An equivalent problem in polar co-ordinates
  3. Chebyshev Alternance of second order on the circle

Key Result

Theorem 1

Let $\dim X =2$. There is a unique $\hat{T}\in\mathbf{PD}(X,\mathbb{R}^2)$ such that $\hat{T}(B_X)\supset B_{\mathbb{R}^2}$ and $\hat{T}$ is characterised by the following property: There are $x_i\in S_X$, $i=1,2$, with $x_1\neq \pm x_2$, and $y_i\in S_X$, $i=1,2$, with $y_1\neq \pm y_2$, such that:

Theorems & Definitions (17)

  • Theorem 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • proof
  • Lemma 1
  • proof
  • Proposition 4
  • proof
  • Proposition 5
  • ...and 7 more