I have a question regarding convexity of logdet function.
Given the convex set $$\mathcal{C}=\left\{{\bf V}_k| {\bf V}_k={\rm diag}\{v_{k,1},...,v_{k,F}\}, \; v_{k,f}\in \mathbb{R}^+,\, \sum_{f=1}^{F} v_{k,f}\leq1\,,\,k=1, \ldots, K\right\}. $$ The function is given as $$ I({\bf V})=\log\,\det\left({\bf I}_F+\sum_{k=1}^{K} {\bf V}_k^{\frac{1}{2}}{{\bf {g}}_k{\bf g}_k^H}{\bf V}_k^{\frac{1}{2}}\right) $$ where ${\bf g}_k\in\mathbb{C}^F$ is a vector given for each index $k$. Is the function $I({\bf V})$ convex/concave in set $\mathcal{C}$?