The Gaussian Minkowski problem is generalized to epigraphs of convex functions, and existence of convex functions realizing prescribed Gaussian moment measures is established under mild conditions.
First variation of functional Wulff shapes
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abstract
We introduce functional Wulff shapes based on the classical construction for compact convex sets. With this new tool, we establish a functional version of Aleksandrov's variational lemma in the family of convex functions with compact domain. The resulting formula is then applied to evaluate the first variation of a class of functionals on convex functions. In particular, we extend a recent result by Huang, Liu, Xi, and Zhao.
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The Gaussian Minkowski problem for epigraphs of convex functions
The Gaussian Minkowski problem is generalized to epigraphs of convex functions, and existence of convex functions realizing prescribed Gaussian moment measures is established under mild conditions.