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Faltings and Hriljac have proved an arithmetic analogue of Hodge index theorem for arithmetic surfaces,by using the Néron-Tate height of the Jacobian.In this talk,I will explain a new proof of this inequality by coupling of measures on R.Based on this idea,by developing results on the transportation of uniform measures between convex bodies,a relative form of Brunn-Minkowski inequality is established for adelic line bundles on higher dimensional arithmetic varities.