Let
$$S=\alpha X + (1-\alpha)Y$$
where $X$ and $Y$ are $m \times n$ matrices with $m > n$ whose singular values are not larger than 1, i.e., $\sigma_{\max}(X) \leq 1$ and $\sigma_{\max}(Y) \leq 1$, where $\sigma_{\max}(\cdot)$ is the largest singular value of the argument. Assume
$$\det(I+XX^T) > \det(I+YY^T)$$
I have observed that $$\arg \max_{0 \leq \alpha \leq 1} \det \left(I + SS^T \right) = 1$$
However, although it looks simple, I fail to prove this. Any ideas?