A solution to two old problems by Menger concerning angle spaces

TL;DR

This paper extends Menger's concept of angle functions to a broader class of spaces with betweenness and successfully solves two longstanding problems about conformal embeddability into Euclidean spaces.

Contribution

The paper generalizes the definition of angle functions to spaces with betweenness and provides solutions to Menger's two classical problems on conformal embeddability.

Findings

Extended angle function definition to betweenness spaces

Solved Menger's problems on conformal embeddability

Broadened understanding of angle spaces in metric and semi-metric contexts

Abstract

Around 1930, K. Menger expressed his interest in the concept of abstract angle function. He introduced a general definition of this notion for metric and semi-metric spaces. He also proposed two problems concerning conformal embeddability of spaces endowed with an angle function into Euclidean spaces. These problems received attention in later years but only for some particular cases of metric spaces. In this article we first update the definition of angle function to apply to the larger class of spaces with a notion of betweenness, which seem to us a more natural framework. In this new general setting, we solve the two problems proposed by Menger.