Let $P$ be a parallelotope in $\mathbb R^n$, centered in $0$ and $\chi$ the characteristic function of $P$ (equals 1 if the input is in the interior or border of $P$ and 0 elsewhere). Assume also that the vertices of $P$ belong to a given lattice, or simply that they are integer points, and that $\chi$ admits only integer points as inputs.
The goal is to find an algorithm that outputs any vertex of $P$ with the smallest amount of queries to $\chi$. I was wondering if there is any known method or algorithm that do this, or related problem or result.