Lipschitz clustering in normed spaces and barycentric spaces

For every normed space $X$ and integers $1\le k<n$, we construct a Lipschitz retraction from $X(n)$ onto $X(k)$ with constant at most $2+8n(n-1)$, independently of the dimension of $X$. Here $X(n)$ consists of the nonempty subsets of $X$ with at most $n$ points, equipped with the Hausdorff metric. For the consecutive case $k=n-1$, a separate construction gives the sharper asymptotic bound $O(n\log n)$. These results extend to barycentric metric spaces and settle some open problems on Lipschitz retractions.