so that $d_{\ell}>0$, for all $\ell\geq 0$.

Suppose that $v_{0}\in\mathcal{C}_{K}$, that is, all the coefficients $a_{i}^{\ell}$ are strictly positive. In that case, there exists a symplectic form $\omega_{0}$ such that $[\omega_{0}]=v_{0}$. The process produces a sequence $\omega_{\ell}$ of symplectic forms which are, by construction, diffeomorphic to the initial symplectic form $\omega_{0}$ and such that, for at least one index $i\geq 1$, $a_{i}^{(\ell)}:=\left< [\omega_{\ell}] , E_{i}\right>$ is a strictly decreasing sequence of positive numbers. By Taubes results on the equivalence of Seiberg-Witten and Gromov invariants, the Gromov invariant of $E_{i}$ only depends on the underlying smooth structure of $X_{n}$. Thus, the class $E_{i}$ contains an embedded $\omega_{\ell}$-symplectic sphere whose size is $a_{i}^{\ell}$. The diffeomorphism from $(X_{n},\omega_{\ell})$ to $(X_{n},\omega_{0})$ carries this sphere to an embedded symplectic sphere in $(X_{n},\omega_{0})$ of size $a_{i}^{\ell}$. However, the set of symplectic areas of exceptional spheres does not have accumulation points, see for instance [@KKP] Lemma 4.1. Thus, the sequence $a_{i}^{\ell}$, being a decreasing sequence of positive numbers that does not have accumulation points, must be finite.

If $v_{0}$ is in the boundary of $\mathcal{C}_{K}$, then all $\hat{v}_{\ell}$ belong to $\partial\mathcal{C}_{K}$ and there are integers $N\geq 0$ and $2\leq m\leq n-1$ such that, for all $\ell\geq N$, 