Auerbach bases of real ${\ell }_p^n$ spaces

Abstract

We study Auerbach bases of the space ${\ell }_p^n$, and in particular, prove that ${\ell }_p^n$ has only finitely many Auerbach bases for $p>n$ and $1<p<n/(n-1)$. An explicit one-parameter family in ${\ell }_4^4$ shows that the condition $p>n$ cannot in general be weakened to $p\ge n$. Further, we provide an explicit classification of the Auerbach bases in dimension two and three.

arXiv comment8 pages. proofs of earlier version have been corrected and refined