proof that $\log(\det(XX^H))$ is not concave

Question

Is there an elegant way to proof that $\log(\det(XX^H))$ is not concave, with respect to the complex-valued matrix elements of $X$?

Answer 1

If $f:X\mapsto \log\det (X X^H)$ were concave you would have $f(0)\ge(f(X)+f(-X))/2$. This is violated by any $X$ of full rank, for which $\det XX^H > 0$.