A Newton-Okounkov body is a convex body in $\mathbb R^n$ associated to a big divisor $D$ on a projective variety representing the asymptotic behavior of the space of global sections $H^0(X,mD)$ when $m$ goes to infinity. Thus for instance, the volume (in $\mathbb R^n$) 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.
