  K-median and k-means are the two most popular objectives for clustering
algorithms. Despite intensive effort, a good understanding of the
approximability of these objectives, particularly in $\ell_p$-metrics, remains
a major open problem. In this paper, we significantly improve upon the hardness
of approximation factors known in literature for these objectives in
$\ell_p$-metrics.
  We introduce a new hypothesis called the Johnson Coverage Hypothesis (JCH),
which roughly asserts that the well-studied max k-coverage problem on set
systems is hard to approximate to a factor greater than 1-1/e, even when the
membership graph of the set system is a subgraph of the Johnson graph. We then
show that together with generalizations of the embedding techniques introduced
by Cohen-Addad and Karthik (FOCS '19), JCH implies hardness of approximation
results for k-median and k-means in $\ell_p$-metrics for factors which are
close to the ones obtained for general metrics. In particular, assuming JCH we
show that it is hard to approximate the k-means objective:
  $\bullet$ Discrete case: To a factor of 3.94 in the $\ell_1$-metric and to a
factor of 1.73 in the $\ell_2$-metric; this improves upon the previous factor
of 1.56 and 1.17 respectively, obtained under UGC.
  $\bullet$ Continuous case: To a factor of 2.10 in the $\ell_1$-metric and to
a factor of 1.36 in the $\ell_2$-metric; this improves upon the previous factor
of 1.07 in the $\ell_2$-metric obtained under UGC.
  We also obtain similar improvements under JCH for the k-median objective.
Additionally, we prove a weak version of JCH using the work of Dinur et al.
(SICOMP '05) on Hypergraph Vertex Cover, and recover all the results stated
above of Cohen-Addad and Karthik (FOCS '19) to (nearly) the same
inapproximability factors but now under the standard NP$\neq$P assumption
(instead of UGC).
