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The ample cone consists of real numerical divisor classes represented by ample real divisors. It is an open convex cone, and Kleiman's criterion identifies it with the interior of the nef cone.

The space of real numerical divisor classes is $N^1(X{)}_{\mathbb{R}}$. Write any big class as $[D]=[A]+[E]$, with $A$ an ample real divisor and $E$ effective. For a sufficiently small perturbation $\delta \in N^1(X{)}_{\mathbb{R}}$, the class $[A]+\delta$ remains in the open ample cone, so $[D]+\delta$ remains big by part (v). Thus $\mathrm{Big}(X)$ is open. Positive scaling and addition preserve the ample-plus-effective expression, so it is a convex cone.

For $n\ge 1$, fix a very ample divisor $H$. The linear functional $\ell ([D])=D\cdot H^{n-1}$ is strictly positive on every big class: the ample part contributes positively and the effective part nonnegatively. No nonzero linear subspace can be contained in this cone, since it would contain both $v$ and $-v$. In particular the zero class is not big in positive dimension. When $n=0$, $N^1(X{)}_{\mathbb{R}}=0$, so there is no positive-dimensional subspace to consider. Therefore