  We obtain using Schwinger's proper time approach the Casimir-Euler-Heisenberg
effective action of fermion fluctuations for the case of an applied magnetic
field. We implement here the compactification of one space dimension into a
circle through anti-periodic boundary condition. Aside of higher order
non-linear field effects we identify a novel contribution to the vacuum
permeability. These contributions are exceedingly small for normal
electromagnetism due to the smallness of the electron Compton wavelength
compared to the size of the compactified dimension, if we take the latter as
the typical size of laboratory cavities, but their presence is thought
provoking, also considering the context of strong interactions.
