On the Converse of Pr\'{e}kopa's Theorem and Berndtsson's Theorem

Abstract

Given a continuous function $\phi$ defined on a domain $\Omega\subset\mathbb{R}^m\times\mathbb{R}^n$, we show that if a Pr\'ekopa-type result holds for $\phi+\psi$ for any non-negative convex function $\psi$ on $\Omega$, then $\phi$ must be a convex function. Additionally, if the projection of $\Omega$ onto $\mathbb{R}^m$ is convex, then $\overline{\Omega}$ is also convex. This provides a converse of Pr\'ekopa's theorem from convex analysis. We also establish analogous results for Berndtsson's theorem on the plurisubharmonic variation of Bergman kernels, showing that the plurisubharmonicity of weight functions and the pseudoconvexity of domains are necessary conditions in some sense.