  The Minkowski functionals are useful statistics to quantify the morphology of
various random fields. They have been applied to numerous analyses of
geometrical patterns, including various types of cosmic fields, morphological
image processing, etc. In some cases, including cosmological applications,
small deviations from the Gaussianity of the distribution are of fundamental
importance. Analytic formulas for the expectation values of Minkowski
functionals with small non-Gaussianity have been derived in limited cases to
date. We generalize these previous works to derive an analytic expression for
expectation values of Minkowski functionals up to second-order corrections of
non-Gaussianity in a space of general dimensions. The derived formula has
sufficient generality to be applied to any random fields with weak
non-Gaussianity in a statistically homogeneous and isotropic space of any
dimensions.
