A local proof of the dimensional Prékopaʹs theorem

The aim of this paper is to find an expression for second derivative of the function ϕ ( t ) defined by ϕ ( t ) = ( ∫ V φ ( t , x ) − β d x ) − 1 β − n , β ≠ n , where U ⊂ R and V ⊂ R n are open bounded subsets, and φ : U × V → R + is a C 2 -smooth function. As a consequence, this result gives us a direct proof of the dimensional Prékopaʹs theorem based on a local approach.