There is no finitely isometric Krivine's theorem

TL;DR

The paper demonstrates that for all p in (1,∞) except 2, there exist Banach spaces isomorphic to ℓ_p containing finite subsets that are not isometric to any subset of the space, disproving a finite isometric Krivine theorem.

Contribution

It proves that the finite isometric version of Krivine's theorem does not hold for p ≠ 2, showing a fundamental limitation in the isometric embedding of finite subsets.

Findings

Existence of Banach spaces isomorphic to ℓ_p with non-isometric finite subsets

Disproof of the finite isometric Krivine theorem for p ≠ 2

Shows limitations of isometric embeddings in Banach space theory

Abstract

We prove that for every $p\in(1,\infty)$, $p\ne 2$, there exist a Banach space $X$ isomorphic to $\ell_p$ and a finite subset $U$ in $\ell_p$, such that $U$ is not isometric to a subset of $X$. This result shows that the finite isometric version of the Krivine theorem (which would be a strengthening of the Krivine theorem (1976)) does not hold.