Approximate isometries on finite dimensional Banach spaces

Abstract:

A map $T:{{\mathbf {E}}_1} \to {{\mathbf {E}}_2}$ (${{\mathbf {E}}_1},{{\mathbf {E}}_2}$ Banach spaces) is an $\epsilon$-isometry if $|\;||T(X) - T(Y)|| - ||X - Y||\;| \leqslant \epsilon$ whenever $X,Y \in {{\mathbf {E}}_1}$. The problem of uniformly approximating such maps by isometries was first raised by Hyers and Ulam in 1945 and subsequently studied for special infinite dimensional Banach spaces. This question is here broached for the class of finite dimensional Banach spaces. The only positive homogeneous candidate isometry $U$ approximating a given $\epsilon$-isometry $T$ is defined by the formal limit $U(X) = {\lim _{r \to \infty }}{r^{ - 1}}T(rX)$. It is shown that, whenever $T:{\mathbf {E}} \to {\mathbf {E}}$ is a surjective $\epsilon$-isometry and ${\mathbf {E}}$ is a finite dimensional Banach space for which the set of extreme points of the unit ball is totally disconnected, then this limit exists. When ${\mathbf {E}} = \ell _1^k( = k \text {- dimensional}\; {\ell _1})$ a uniform bound of uniform approximation is obtained for surjective $\epsilon$-isometries by isometries; this bound varies linearly in $\epsilon$ and with ${k^3}$.