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A local proof of the dimensional Pr\'ekopa's theorem
The aim of this paper is to find an expression for second derivative of the
function defined by \phi(t) = \lt(\int_V \vphi(t,x)^{-\beta}
dx\rt)^{-\frac1{\be -n}},\qquad \beta\not= n, where and
are open bounded subsets, and \vphi: U\times V\to \R_+ is a
smooth function. As a consequence, this result gives us a direct proof of
the dimensional Pr\'ekopa's theorem based on a local approach.Comment: 9 pages, to appear in J. Math. Anal. App
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