A Determinantal Inequality for the Geometric Mean with an Application in Diffusion Tensor Imaging

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