  We consider the well-studied Robust $(k, z)$-Clustering problem, which
generalizes the classic $k$-Median, $k$-Means, and $k$-Center problems. Given a
constant $z\ge 1$, the input to Robust $(k, z)$-Clustering is a set $P$ of $n$
weighted points in a metric space $(M,\delta)$ and a positive integer $k$.
Further, each point belongs to one (or more) of the $m$ many different groups
$S_1,S_2,\ldots,S_m$. Our goal is to find a set $X$ of $k$ centers such that
$\max_{i \in [m]} \sum_{p \in S_i} w(p) \delta(p,X)^z$ is minimized.
  This problem arises in the domains of robust optimization [Anthony, Goyal,
Gupta, Nagarajan, Math. Oper. Res. 2010] and in algorithmic fairness. For
polynomial time computation, an approximation factor of $O(\log m/\log\log m)$
is known [Makarychev, Vakilian, COLT $2021$], which is tight under a plausible
complexity assumption even in the line metrics. For FPT time, there is a
$(3^z+\epsilon)$-approximation algorithm, which is tight under GAP-ETH [Goyal,
Jaiswal, Inf. Proc. Letters, 2023].
  Motivated by the tight lower bounds for general discrete metrics, we focus on
\emph{geometric} spaces such as the (discrete) high-dimensional Euclidean
setting and metrics of low doubling dimension, which play an important role in
data analysis applications. First, for a universal constant $\eta_0 >0.0006$,
we devise a $3^z(1-\eta_{0})$-factor FPT approximation algorithm for discrete
high-dimensional Euclidean spaces thereby bypassing the lower bound for general
metrics. We complement this result by showing that even the special case of
$k$-Center in dimension $\Theta(\log n)$ is $(\sqrt{3/2}- o(1))$-hard to
approximate for FPT algorithms. Finally, we complete the FPT approximation
landscape by designing an FPT $(1+\epsilon)$-approximation scheme (EPAS) for
the metric of sub-logarithmic doubling dimension.
