  We use a locally constant field approximation (LCFA) to study the one-loop
Heisenberg-Euler effective action in a particular class of slowly varying
inhomogeneous electric fields of Lorentzian shape with $0\leq d\leq 4$
inhomogeneous directions. We show that for these fields, the LCFA of the
Heisenberg-Euler effective action can be represented in terms of a single
parameter integral, with the constant field effective Lagrangian with rescaled
argument as integration kernel. The imaginary part of the Heisenberg-Euler
effective action contains information about the instability of the quantum
vacuum towards the formation of a state with real electrons and positrons.
Here, we in particular focus on the dependence of the instantaneous vacuum
decay rate on the dimension $d$ of the field inhomogeneity. Specifically for
weak fields, we find an overall parametric suppression of the effect with
$(E_0/E_{\rm cr})^{d/2}$, where $E_0$ is the peak field strength of the
inhomogeneity and $E_{\rm cr}$ the critical electric field strength.
