Banach-Mazur distance and B-convex Banach spaces(The structure of Banach spaces and Function spaces)

A Banach space $X$ is said to be $\mathrm{B}$ -convex if it is $B_{n}$ -convex for some $n\geq 2$ .As is well-known, $\mathrm{B}$ -convexity is an isomorphic invariant, but $B_{n}$ -convexity is not so.In this short note, we are concerned with the stability of $\mathrm{B}_{n}$ -convexity under norm perturvations.It is known (cf.[7]) that $X$ is, where $d(X, \mathrm{Y})$ denotes the Banach-Mazur distance between $X$ and $\mathrm{Y}$ ; and this implies that if $X$ is $B_{n^{-}}$ convex, then there exists $\mathrm{A}_{n}>1$ such that all Banach spaces $\mathrm{Y}$ satisfying $d(X, \mathrm{Y})<\lambda_{n}$ are $\mathrm{B}_{n}$ -convex.In the case $X=l_{p}$ or $L_{\mathrm{p}}[0,1],$ $1<p<\infty$ , it is also shown that all Banach spaces $\mathrm{Y}$ satisfying $d(X, \mathrm{Y})