  Tests based on the $2$- and $\infty$-norm have received considerable
attention in high-dimensional testing problems, as they are powerful against
dense and sparse alternatives, respectively. The power enhancement principle of
Fan et al. (2015) combines these two norms to construct tests that are powerful
against both types of alternatives. Nevertheless, the $2$- and $\infty$-norm
are just two out of the whole spectrum of $p$-norms that one can base a test
on. In the context of testing whether a candidate parameter satisfies a large
number of moment equalities, we construct a test that harnesses the strength of
all $p$-norms with $p\in[2, \infty]$. As a result, this test consistent against
strictly more alternatives than any test based on a single $p$-norm. In
particular, our test is consistent against more alternatives than tests based
on the $2$- and $\infty$-norm, which is what most implementations of the power
enhancement principle target.
  We illustrate the scope of our general results by using them to construct a
test that simultaneously dominates the Anderson-Rubin test (based on $p=2$) and
tests based on the $\infty$-norm in terms of consistency in the linear
instrumental variable model with many (weak) instruments.
