Properties of the Log Determinant

It is well known that \( f(\Xv) = -\log\det \Xv \) is a convex function on the positive definite cone \( \Sb_{+ +}^d \).

1. Show that \(h(\Xv) = \Xv\inv\) defined on \(\Sb_{+ +}^d\) is \(1\)-Lipschitz in the spectral norm when \(\Xv \succeq \Iv\).

2. Redo (1) with the Frobenius norm.

3. Show that \( f(\Xv) \) is \( 1 \)-smooth in the Frobenius norm when \(\Xv \succeq \Iv\).

4. Show that \( f(\Xv) \) is \( 1 \)-strongly convex in the Frobenius norm when \(\Xv \preceq \Iv\).

5. Show that \( g(\zv, \Xv) = -\log\det(\Xv - \zv \zv^\top) \) is jointly convex on the domain \( \Xv - \zv \zv^\top \succ 0 \).
   Note: This is the convex conjugate of the log-partition function of Gaussian distributions, but you have to do the proof without resorting to the conjugacy.