  This article is concerned with the asymptotic behavior of certain sequences
of ideals in rings of prime characteristic. These sequences, which we call
$p$-families of ideals, are ubiquitous in prime characteristic commutative
algebra (e.g., they occur naturally in the theories of tight closure,
Hilbert-Kunz multiplicity, and $F$-signature). We associate to each $p$-family
of ideals an object in Euclidean space that is analogous to the Newton-Okounkov
body of a graded family of ideals, which we call a $p$-body. Generalizing the
methods used to establish volume formulas for the Hilbert-Kunz multiplicity and
$F$-signature of semigroup rings, we relate the volume of a $p$-body to a
certain asymptotic invariant determined by the corresponding $p$-family of
ideals. We apply these methods to obtain new existence results for limits in
positive characteristic, an analogue of the Brunn-Minkowski theorem for
Hilbert-Kunz multiplicity, and a uniformity result concerning the positivity of
a $p$-family.
