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

TL;DR

This paper explores the converse of Prékopa's and Berndtsson's theorems, demonstrating that certain convexity and plurisubharmonicity conditions are necessary for these theorems to hold, thus deepening understanding of their foundational assumptions.

Contribution

It establishes the necessity of convexity and pseudoconvexity conditions for Prékopa-type and Berndtsson-type theorems, providing converse results that clarify their underlying assumptions.

Findings

Convexity of $ar{ ext{domain}}$ follows from Prékopa-type results.

Convexity of the domain's projection implies convexity of the domain itself.

Plurisubharmonicity of weights is necessary for 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.