On the Converse of Prékopa's Theorem and Berndtsson's Theorem
Wang Xu, Hui Yang
TL;DR
The paper develops converse results to two cornerstone theorems in convex and complex analysis: Prékopa's inequality and Berndtsson's plurisubharmonic variation of Bergman kernels. By introducing twisted convex and psh weights ψ, the authors show that if a Prékopa-type marginal is convex for all nonnegative convex ψ, then the base function φ must be convex, and similarly, if the fiberwise Bergman kernel variation preserves log-pshity under all bounded-below psh ψ, then φ must be psh. The work also establishes necessary geometric hypotheses, proving that convexity of the base projection implies convexity of the closure of the domain, and that pseudoconvexity (with mild regularity) is necessary for Berndtsson-type results. A series of counterexamples clarifies the sharpness of these twists and regularity assumptions, and a parallel result links the pseudoconvexity of the domain to the fiberwise Bergman kernel behavior. The results broaden the understanding of when convexity/psh-variation conclusions must originate from the underlying data and domain geometry.
Abstract
Given a continuous function $φ$ defined on a domain $Ω\subset\mathbb{R}^m\times\mathbb{R}^n$, we show that if a Prékopa-type result holds for $φ+ψ$ for any non-negative convex function $ψ$ on $Ω$, then $φ$ must be a convex function. Additionally, if the projection of $Ω$ onto $\mathbb{R}^m$ is convex, then $\overlineΩ$ is also convex. This provides a converse of Prékopa'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.