  The $k$-Center problem is one of the most popular clustering problems. After
decades of work, the complexity of most of its variants on general metrics is
now well understood. Surprisingly, this is not the case for a natural setting
that often arises in practice, namely the Euclidean setting, in which the input
points are points in $\mathbb{R}^d$, and the distance between them is the
standard $\ell_2$ Euclidean distance. In this work, we study two Euclidean
$k$-Center variants, the Matroid Center problem on the real line and the Robust
Euclidean $k$-Supplier problem, and provide algorithms that improve upon the
best approximation guarantees known for these problems. In particular, we
present a simple $2.5$-approximation algorithm for the Matroid Center problem
on the real line, thus improving upon the $3$-approximation factor algorithm of
Chen, Li, Liang, and Wang (2016) that works for general metrics. Moreover, we
present a $(1 + \sqrt{3})$-approximation algorithm for the Robust Euclidean
$k$-Supplier problem, thus improving upon the state-of-the-art
$3$-approximation algorithm for Robust $k$-Supplier on general metrics and
matching the best approximation factor known for the non-robust setting by
Nagarajan, Schieber and Shachnai (2020).
