On Berndtsson's generalization of Prekopa's theorem
Résumé
We present a new approach to Berndtsson's complex extension of Prekopa's theorem. This approach is inspired by the recent local proof of Prekopa's theorem obtained by Ball, Barthe and Naor. In the complex setting, this approach leads to a formula expressing DeltaPhi in terms of (derivatives of)omega where e(-Phi(z)) = integrale(-phi(z,w)) dlambda(w) As a consequence, we obtain new conditions ensuring that Phi is subharmonic when phi is plurisubharmonic.