Inequalities and counterexamples for functional intrinsic volumes and beyond

We show that analytic analogs of Brunn-Minkowski-type inequalities fail for functional intrinsic volumes on convex functions. This is demonstrated both through counterexamples and by connecting the problem to results of Colesanti, Hug, and Saorín Gómez. By restricting to a smaller set of admissible functions, we then introduce a family of variational functionals and establish Wulff-type inequalities for these quantities. In addition, we derive inequalities for the corresponding family of mixed functionals, thereby generalizing an earlier Alexandrov-Fenchel-type inequality by Klartag and recovering a special case of a recent Pólya-Szegő-type inequality by Bianchi, Cianchi, and Gronchi.