Summary: Unlike the case in one dimension, there is still much to learn about the basic nature of the family \( \mathcal {K}(\mathbb{B})\) of normalized (\(f(0)=0\), \(Df(0)=I\), where \(Df\) is the Fréchet derivative of \(f\) and \(I\) is the identity operator on \( \mathbb{C}^n\)) biholomorphic mappings \(f\) of the Euclidean unit ball \( \mathbb{B} \subseteq \mathbb{C}^n\) onto convex domains in \( \mathbb{C}^n\) when \( n\geq 2\). We consider its closed convex hull \( \overline {\mathrm{co}}\, \mathcal {K}(\mathbb{B})\) in relation to the family \( \mathcal {R}(\mathbb{B})\) of normalized holomorphic mappings \( f\colon \mathbb{B} \rightarrow \mathbb{C}^n\) satisfying \( \operatorname {Re} \langle f(z),z \rangle > \Vert z\Vert^2/2\) for \( z\in \mathbb{B} \setminus \{0\}\), where \( \langle \cdot ,\cdot \rangle \) and \( \Vert\cdot \Vert\) are, respectively, the Hermitian inner product and Euclidean norm in \( \mathbb{C}^n\). In dimension \(n=1\), the sets are the same. Here, we identify some extreme points of \( \overline {\text{co}}\, \mathcal {K}(\mathbb{B})\) and use them to show that \( \overline {\text{co}}\, \mathcal {K}(\mathbb{B})\) is a proper subset of \( \mathcal {R}(\mathbb{B})\) when \( n\geq 2\). We also consider an extension operator related to \( \mathcal {R}(\mathbb{B})\) that helps illustrate where the known extreme points of \( \overline {\text{co}}\, \mathcal {K}(\mathbb{B})\) lie in \( \mathcal {R}(\mathbb{B})\) and make some observations on the related case of the unit polydisk.