A VOLUME = MULTIPLICITY FORMULA FOR p-FAMILIES OF IDEALS

. In this article, we work with certain families of ideals called pfamilies in rings of prime characteristic. This family of ideals is present in the theories of tight closure, Hilbert-Kunz multiplicity, and F-signature. For each p-family of ideals, we attach an Euclidean object called p-body, which is analogous to the Newton Okounkov body associated with a graded family of ideals. Using the combinatorial properties of p-bodies and algebraic properties of the Hilbert-Kunz multiplicity, we establish a Volume = Multiplicity formula for p-families of mR-primary ideals in a Noetherian local ring R.