A local proof of the dimensional Prékopa's theorem

Abstract

The aim of this paper is to find an expression for second derivative of the function $\varphi (t)$ defined by $$ϕ(t) = \lt(\int_V \vphi(t,x)^{-β} dx\rt)^{-\frac1{\be -n}},\qquad β\not= n,$$ where $U\subset \mathbb{R}$ and $V\subset {\mathbb{R}}^n$ are open bounded subsets, and $\vphi: U\times V\to \R_+$ is a $C^2-$smooth function. As a consequence, this result gives us a direct proof of the dimensional Prékopa's theorem based on a local approach.

9 pages, to appear in J. Math. Anal. Appl