A quest for 5-point condition a la Alexandrov

TL;DR

This paper explores a new 5-point geometric condition inspired by Alexandrov's 4-point comparison, aiming to advance understanding in metric geometry.

Contribution

It introduces a novel 5-point condition related to Alexandrov geometry, expanding the framework for geometric comparison theorems.

Findings

Proposes a 5-point condition as a potential extension of Alexandrov's 4-point comparison.

Provides a quadratic form-based description of Alexandrov's 4-point comparison.

Suggests future research directions for the 5-point condition.

Abstract

Here I give a description of Alexandrov 4-point comparison via quadratic forms and then propose a natural 5-point condition which might have some future. Consider this note as a letter from me --- do not take it seriously.