Equilateral sets in uniformly smooth Banach spaces

TL;DR

This paper proves that every infinite dimensional uniformly smooth Banach space contains an infinite set of points all equally distant from each other, expanding understanding of geometric structures in such spaces.

Contribution

It establishes the existence of infinite equilateral sets in all infinite dimensional uniformly smooth Banach spaces, a previously unresolved geometric property.

Findings

Existence of infinite equilateral sets in uniformly smooth Banach spaces

Extension of geometric structure understanding in Banach space theory

Advancement in the study of metric properties of Banach spaces

Abstract

Let $X$ be an infinite dimensional uniformly smooth Banach space. We prove that $X$ contains an infinite equilateral set. That is, there exists a constant $\lambda>0$ and an infinite sequence $(x_i)_{i=1}^\infty\subset X$ such that $\|x_i-x_j\|=\lambda$ for all $i\neq j$.