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 S^n-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.