In joint work with Malabika Pramanik, and motivated by problems in complex analysis, we consider maximal averages over sets in $\mathbb R^{n}$ defined by a finite number $d\geq n$ of monomial inequalities. Fixing the monomials but varying the inequalities gives us a $d$-parameter family of sets, which we think of as a family of balls. If $d=n$ and the monomials are just the coordinate functions, then these sets are all rectangles with sides parallel to the axes. If $d>>n$, the structure of monomial polyhedra can be rather complicated.
We shall discuss the geometry of monomial polyhedra and the associated maximal operator. In general, this operator is larger than the strong maximal function introduced by Jessen, Marcinkiewicz, and Zygmund. Nevertheless we show that under suitable hypotheses, this operator is bounded on $L^{p}$ for $1 < p \leq \infty$, and has the same behavior near $L^{1}$ as the strong maximal function. We shall also briefly discuss connections to estimates for the Bergman kernel in certain domains in several complex variables.