Convex hull of compact set is compact in finite dimensional complex vector spaces. I am wondering whether Caratheodory's convexity theorem extends to finite dimensional vector spaces over the complex numbers.
Really, I am trying to prove that the convex hull of the set of projections in $M_n(\mathbb{C})$ (when equipped as a C*-algebra) is equal to the positive unit ball. Therefore, having the complex equivalent of Caratheodory together with Krein-Milman would establish the result. 
Am I overlooking an easy exit to this problem?
Thank you.
 A: To answer your question, you can think of $\mathbb C^n$ as $\mathbb R^{2n}$ and apply Caratheodory. 
Now, there is a straightforward proof. Given $A\in M_n(\mathbb C)$, positive with $\|A\|\leq1$, by the Spectral Theorem we can write $$A=\sum_{j=1}^r\alpha_jP_j,$$ where $1\geq\alpha_1\geq\cdots\geq\alpha_r>0$, with $r\leq n$, and $P_1,\ldots,P_n$ are pairwise orthogonal. We may write (putting $\alpha_{r+1}=\alpha_0=0$)
\begin{align}
A&=\alpha_r\sum_{j=1}^r P_j+\sum_{j=1}^{r-1}(\alpha_j-\alpha_r)P_j\\[0.3cm]
&=\alpha_r\sum_{j=1}^r P_j+(\alpha_{r-1}-\alpha_r)\sum_{j=1}^{r-1}P_j+\sum_{j=1}^{r-2}(\alpha_j-\alpha_{r-1})P_j\\[0.3cm]
&=\alpha_r\sum_{j=1}^r P_j+(\alpha_{r-1}-\alpha_r)\sum_{j=1}^{r-1}P_j+(\alpha_{r-2}-\alpha_{r-1})\sum_{j=1}^{r-2}P_j+\sum_{j=1}^{r-3}(\alpha_j-\alpha_{r-2})P_j\\[0.3cm]
&=\sum_{k=0}^{r-1}(\alpha_{r-k}-\alpha_{r-k+1})\sum_{j=1}^{r-k}P_j.
\end{align}
Since $\sum_{k=0}^{r-1}(\alpha_{r-k}-\alpha_{r-k+1})=\alpha_r$, we have 
$$\tag1
A=(1-\alpha_r)\,0+\sum_{k=0}^{r-1}(\alpha_{r-k}-\alpha_{r-k+1})Q_k,
$$
where $Q_k=\sum_{j=1}^{r-k}P_j$ is a projection since it is a sum of pairwise orthogonal projections. Then $(1)$ expresses $A$ as a convex combination of projections. 
