  In this article, we study shape fitting problems, $\epsilon$-coresets, and
total sensitivity. We focus on the $(j,k)$-projective clustering problems,
including $k$-median/$k$-means, $k$-line clustering, $j$-subspace
approximation, and the integer $(j,k)$-projective clustering problem. We derive
upper bounds of total sensitivities for these problems, and obtain
$\epsilon$-coresets using these upper bounds. Using a dimension-reduction type
argument, we are able to greatly simplify earlier results on total sensitivity
for the $k$-median/$k$-means clustering problems, and obtain
positively-weighted $\epsilon$-coresets for several variants of the
$(j,k)$-projective clustering problem. We also extend an earlier result on
$\epsilon$-coresets for the integer $(j,k)$-projective clustering problem in
fixed dimension to the case of high dimension.
