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I want to know the following: If $x_1, x_2, \cdots, x_n, y_1,y_2, \cdots, y_n \in \ell_p$ satisfies $\|x_i-x_j\|_p=\|y_i-y_j\|_p$ for any $i,j$, then does there exist isometry $F$ of $\ell_p$ which send each $x_i$ to $y_i$ ? Also do you know the precise description of the isometry group of $\ell_p$ ? |
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To answer your last question, you must combine two classical results: 1) First, the Mazur-Ulam theorem tells you that every (surjective) isometry of a real Banach space, is affine. This reduces the description of the isometry group, to the group of linear isometries. 2) By a corollary of the Banach-Lamperti theorem, every linear isometry $T$ of $\ell^p=\ell^p(\mathbb{N})$ (with $1\leq p<\infty,p\neq 2$) is of the form $T:(a_n)\mapsto (\epsilon(n)a_{\sigma(n)})$, where $\sigma$ is a permutation of $\mathbb{N}$, and $\epsilon(n)=\pm 1$ for every $n$. |
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The answer to the first question is NO. Even among norms on $\mathbb R^2$, the only ones that have this amazing property (any isometry defined on a finite set extends to an isometry defined on the whole space) are those norms that make $\mathbb R^2$ isometrically into Euclidean space. |
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