Differentiability of volumes of divisors and a problem of teissier

We give an algebraic construction of the positive intersection products of pseudo-effective classes and use them to prove that the volume function on the Néron-Severi space of a projective variety is t¹- differentiable, expressing its differential as a positive intersection product. We also relate the differential to the restricted volumes. We then apply our differentiability result to prove an algebro-geometric version of the Diskant inequality in convex geometry, allowing us to characterize the equality case of the Khovanskii-Teissier inequalities for nef and big classes.