  In this paper, we introduce positivity notions for pairs of adelic R-Cartier
divisors and R-base conditions, and study fundamental properties of the
arithmetic volumes defined for such pairs. We show that the Gateaux derivatives
of the arithmetic volume function at big pairs along the directions of adelic
R-Cartier divisors are given by suitable arithmetic positive intersection
numbers. As a corollary, we obtain an Arakelov theoretic analogue of the
Bonnesen-Diskant inequality in convex geometry.
