A Newton-Okounkov body is a convex body in R^n associated to a big divisor D on a projective variety representing the asymptotic bahavior of the space of global section H^0(X,mD) when m goes to infinity. Thus for instance, the volume (in R^n) of the Newton-Okounkov body of D is n! times the volume of the divisor D. Lehmann and Xiao have defined dual notions of volume for curves (instead of divisors). We will see that it is possible to construct Newton-Okounkov bodies for curves whose volume is n! times the volume of the initial curve. Additionally this construction allows us to establish a new conjecture on Newton-Okounkov bodies.
