# A note on symmetric separation in Banach spaces

## TL;DR

This paper establishes that the symmetric Kottman's constant exceeds 1 in all infinite-dimensional Banach spaces and explores its behavior in spaces with $c_0$ spreading models, advancing understanding of geometric properties of Banach spaces.

## Contribution

It proves that $K^s(X)>1$ for all infinite-dimensional Banach spaces and investigates the constant in spaces with $c_0$ spreading models, solving open problems.

## Key findings

- $K^s(X)>1$ for every infinite-dimensional Banach space
- Characterization of $K^s(X)$ in spaces with $c_0$ spreading models
- Answered an open question from previous research

## Abstract

We present some new results on the symmetric Kottman's constant $K^s(X)$ of a Banach space $X$ and its relationship with the Kottman constant. We show that $K^s(X)>1$, for every infinite-dimensional Banach space, thereby solving a problem by J.M.F. Castillo and P.L. Papini. We also investigate such constant in the class of Banach spaces admitting $c_0$ spreading models, answering in particular one question from our previous joint paper with P. H\'ajek and T. Kania.