Okounkov bodies and restricted volumes along very general curves

Given a big divisor D on a normal complex projective variety X, we show that the restricted volume of D along a very general complete-intersection curve C⊂X can be read off from the Okounkov body of D with respect to an admissible flag containing C. From this we deduce that if two big divisors D1 and D2 on X have the same Okounkov body with respect to every admissible flag, then D1 and D2 are numerically equivalent.