The Gaussian Minkowski problem for epigraphs of convex functions

A variational formula is derived by combining the Gaussian volume of the epigraph of a convex function $φ$ and the perturbation of $φ$ via the infimal convolution. This formula naturally leads to a Borel measure on Rⁿ and a Borel measure on the unit sphere Sⁿ⁻¹. The resulting Borel measure on Rⁿ will be called the Euclidean Gaussian moment measure of the convex function $φ$, and the related Minkowski-type problem will be studied. In particular, the newly posed Minkowski problem is solved under some mild and natural conditions on the pre-given measure.