  We establish $L^{p_1}\times\cdots\times L^{p_k}\to L^r$ and
$\ell^{p_1}\times\cdots\times \ell^{p_k}\to \ell^r$ type bounds for multilinear
maximal operators associated to averages over isometric copies of a given
non-degenerate $k$-simplex in both the continuous and discrete settings. These
provide natural extensions of $L^p\to L^p$ and $\ell^p\to \ell^p$ bounds for
Stein's spherical maximal operator and the discrete spherical maximal operator,
with each of these results serving as a key ingredient of the respective
proofs.
