Discrete multilinear maximal operators associated to simplices

We prove $\ell^{p_1}\times\cdots\times \ell^{p_k}\to \ell^r$ bounds for multilinear maximal operators associated to averages over all isometric copies of a given non-degenerate $k$-simplex. This provides a natural extension of $\ell^p\to \ell^p$ bounds for the discrete spherical maximal operator, which also serves as the key ingredient of our proof.