  We show that the base polytope $P_M$ of any paving matroid $M$ can be
systematically obtained from a hypersimplex by slicing off certain
subpolytopes, namely base polytopes of lattice path matroids corresponding to
panhandle-shaped Ferrers diagrams. We calculate the Ehrhart polynomials of
these matroids and consequently write down the Ehrhart polynomial of $P_M$,
starting with Katzman's formula for the Ehrhart polynomial of a hypersimplex.
The method builds on and generalizes Ferroni's work on sparse paving matroids.
Combinatorially, our construction corresponds to constructing a uniform matroid
from a paving matroid by iterating the operation of stressed-hyperplane
relaxation introduced by Ferroni, Nasr, and Vecchi, which generalizes the
standard matroid-theoretic notion of circuit-hyperplane relaxation. We present
evidence that panhandle matroids are Ehrhart positive and describe a
conjectured combinatorial formula involving chain forests and Eulerian numbers
from which Ehrhart positivity of panhandle matroids will follow. As an
application of the main result, we calculate the Ehrhart polynomials of
matroids associated with Steiner systems and finite projective planes, and show
that they depend only on their design-theoretic parameters: for example, while
projective planes of the same order need not have isomorphic matroids, their
base polytopes must be Ehrhart equivalent.
