On Minkowski decomposition of Okounkov bodies on a Del Pezzo surface

We show that on a blow up of $P^2$ in $3$ general points there exists a finite set of nef divisors $P_1,ldots,P_s$ such that the Okounkov body $Delta(D)$ of an arbitrary effective $R$--divisor $D$ on $X$ is the Minkowski sum Delta(D)=sum_{i=1}^s a_iDelta(P_i) (1) with non-negative coefficients $a_i in R_{geq 0}$.