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A characterisation of Euclidean normed planes via bisectors

Javier Cabello Sánchez, Adrián Gordillo-Merino

Abstract

Our main result states that whenever we have a non-Euclidean norm $\|\cdot\|$ on a two-dimensional vector space $X$, there exists some $x\neq 0$ such that for every $λ\neq 1, λ>0$, there exist $y, z\in X$ verifying that $\|y\|=λ\|x\|$, $z\neq 0$, and $z$ belongs to the bisectors $B(-x,x)$ and $B(-y,y)$. Throughout this paper we also state and prove some other simple but maybe useful results about the geometry of the unit sphere of strictly convex planes.

A characterisation of Euclidean normed planes via bisectors

Abstract

Our main result states that whenever we have a non-Euclidean norm on a two-dimensional vector space , there exists some such that for every , there exist verifying that , , and belongs to the bisectors and . Throughout this paper we also state and prove some other simple but maybe useful results about the geometry of the unit sphere of strictly convex planes.
Paper Structure (4 sections, 9 theorems, 38 equations)

This paper contains 4 sections, 9 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Side results
  3. Proof of the other implication
  4. Main result

Key Result

Proposition 1

A norm $\|\cdot\|$ on $X\,$ is strictly convex if and only if for every nonzero $z\in X\,$ there exists, up to $\pm 1$, exactly one vector which is isosceles orthogonal to $z$ in $S_X$.

Theorems & Definitions (20)

  • Proposition
  • Conjecture
  • Theorem
  • Remark 1.1
  • Proposition 2.1
  • Remark 2.2
  • Remark 2.3
  • proof : Proof of the easy implication of Prop. \ref{['anticonjetura']}
  • Lemma 2.4
  • proof
  • ...and 10 more