Determinant inequality in terms of operator norm

Question

Let A,B real symmetric matrices $2n \times 2n$ both positive semidefinite. How can I upper bound $\mathrm{Det}(\frac{A+B}{2})-\sqrt{\mathrm{Det}(AB)}$ in terms of the operator norm difference of the two matrices $\| A-B\|$? Here the operator norm of a matrix is its largest singular value.

Answer 1

You can't bound it in terms of $\|A-B\|$ alone. Consider the case $n=2$ (i.e. $4 \times 4$ matrices) with $B = t I$ and $A = (t+1) I$. Then

$$ \text{Det}\left(\frac{A+B}{2}\right) - \sqrt{\text{Det}(AB)} = \frac{t^2}{2} + \frac{t}{2} + \frac{1}{16}$$

while $\|A-B\| = 1$.