Prekopa's theorem and Kiselman's minimum principle for plurisubharmonic functions (Q1298118)

The minimum principle for convex functions states that if \(\varphi(x, y)\) is a convex function then the marginal function \(\varphi^\ast(x) \) defined as the infimum over \(y\) is also convex. \textit{A. Prekopa} [Acad. Sci. Math. 34, 335-343 (1973; Zbl 0264.90038)] found an integral version of the minimum principle: The function \(\widetilde\varphi(x)\) defined by \(-\log \int e^{-\varphi(x, y)} dy\) is also convex if \(\varphi(x, y)\) is convex. A generalization of the minimum principle to plurisubharmonic functions was given by \textit{C. O. Kiselman} [Invent. Math. 49, No. 2, 137-148 (1978; Zbl 0388.32009)], which asserts, among other things, that the marginal function \(\varphi^\ast(z) \) of a plurisubharmonic function \(\varphi(z, w)\) on \(U\times V \) is also plurisubharmonic, if \(V\) is a pseudoconvex Reinhard domain and \(\varphi(z, w)\) is independent of the arguments \(\arg(w_j)\) of \(w=(w_1, \dots, w_m)\). In the present paper the author generalizes Kiselman's result and proves an integral version of the result.