  We obtain the first strong coresets for the $k$-median and subspace
approximation problems with sum of distances objective function, on $n$ points
in $d$ dimensions, with a number of weighted points that is independent of both
$n$ and $d$; namely, our coresets have size $\text{poly}(k/\epsilon)$. A strong
coreset $(1+\epsilon)$-approximates the cost function for all possible sets of
centers simultaneously. We also give efficient $\text{nnz}(A) +
(n+d)\text{poly}(k/\epsilon) + \exp(\text{poly}(k/\epsilon))$ time algorithms
for computing these coresets.
  We obtain the result by introducing a new dimensionality reduction technique
for coresets that significantly generalizes an earlier result of Feldman,
Sohler and Schmidt \cite{FSS13} for squared Euclidean distances to sums of
$p$-th powers of Euclidean distances for constant $p\ge1$.
