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Let 
${{\mathcal{H}}_{n}}$ be the real linear space of 
$n\,\times \,n$ complex Hermitian matrices. The unitary (similarity) orbit 
$\mathcal{U}\left( C \right)$ of 
$C\,\in \,{{\mathcal{H}}_{n}}$ is the collection of all matrices unitarily similar to 
$C$. We characterize those $C\,\in \,{{\mathcal{H}}_{n}}$ such that every matrix in the convex hull of $\mathcal{U}\left( C \right)$ can be written as the average of two matrices in $\mathcal{U}\left( C \right)$. The result is used to study spectral properties of submatrices of matrices in $\mathcal{U}\left( C \right)$, the convexity of images of $\mathcal{U}\left( C \right)$ under linear transformations, and some related questions concerning the joint $C$-numerical range of Hermitian matrices. Analogous results on real symmetric matrices are also discussed.
