  Let $S_n$ be the symmetric group on the set $\{1,2,\ldots,n\}$. Given a
permutation $\sigma=\sigma_1\sigma_2 \cdots \sigma_n \in S_n$, we say it has a
peak at index $i$ if $\sigma_{i-1}<\sigma_i>\sigma_{i+1}$. Let
$\text{Peak}(\sigma)$ be the set of all peaks of $\sigma$ and define
$P(S;n)=\{\sigma\in S_n\, | \,\text{Peak}(\sigma)=S\}$. In this paper we study
the Hamming metric, $\ell_\infty$-metric, and Kendall-Tau metric on the sets
$P(S;n)$ for all possible $S$, and determine the minimum and maximum possible
values that these metrics can attain in these subsets of $S_n$.
