On the Pelczynski conjecture on Auerbach Bases

TL;DR

This paper investigates the number of Auerbach bases in finite-dimensional Banach spaces, providing lower bounds and refined estimates using topological and Morse theory methods.

Contribution

It establishes new lower bounds on the number of Auerbach bases and improves estimates for generic spaces through advanced topological techniques.

Findings

At least (n-1)n/2+1 Auerbach bases exist in n-dimensional Banach spaces.

Lusternik-Schnirelmann category calculation yields initial bounds.

Morse theory provides better estimates for generic Banach spaces.

Abstract

We consider Auerbach bases in Banach spaces of dimension n>2. We show that there exists at least (n-1)n/2+1 such bases. This estimate follows from the calculation of the Lusternik-Schnirelmann category of the flag variety. A better estimate is obtained for generic Banach spaces by the Morse theory.