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The volume inequalities for general  zonoids of even isotropic measures and for their duals are strengthened by Ball et al.. Motivated by their way, a stronger version of the Brascamp-Lieb inequality for a family of functions is proved, which can approximate arbitrary well some Gaussians when equality holds. Its application gives the  Loomis-Whitney inequality for even isotropic measures associated with the support function of  projection bodies with complete equality conditions. Moreover, we establish a dual version of the Loomis-Whitney inequality for isotropic measures with complete equality conditions, in which we give the sharp lower bound for the volumes of hyperplane sections. This extends Ball’s Loomis-Whitney inequality and dual Ball’s Loomis-Whitney inequality to the  space, respectively. 

volume, isoperimetric inequality, reverse isoperimetric inequality, general -zonoid, Brascamp-Lieb inequality, mass transportation, stability result, isotropic measure,  Loomis-Whitney inequality.