$F$-pure thresholds and $F$-Volumes of some non principal ideals

We compute the $F$-pure threshold of some non-principal ideals which satisfy a geometric generic condition about their Newton polyhedron. We also contribute some evidence in favor of the conjectured equality between the $F$-pure threshold and the log canonical threshold of ideals for infinitely many primes $p$. These results are obtained by generalizing the theory of splitting polytopes to the case of ideals. As an application of our results we obtain geometric lower bounds for the recently introduced $F$-volume of a collection of ideals.