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 $\phi$ and the perturbation of $\phi$ via the infimal convolution. This formula naturally leads to a Borel measure on ${\mathbb{R}}^n$ and a Borel measure on the unit sphere $S^{n-1}$. The resulting Borel measure on ${\mathbb{R}}^n$ will be called the Euclidean Gaussian moment measure of the convex function $\phi$, 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.