  We prove that for positive semidefinite matrices $A$ and $B$ the following
determinantal inequality holds: \[ \det(I+A\#B)\le \det(I+A^{1/2}B^{1/2}), \]
where $A\#B$ is the geometric mean of $A$ and $B$. We apply this inequality to
the study of interpolation methods in diffusion tensor imaging.
