Suppose I have a diagonal $n \times n$ matrix $\Gamma$ with positive entries, and a fixed $n \times k$ matrix $P$ with $P^\intercal P = I$ (here, $k \leq n$). I'm interested in knowing whether the function
$$ f(\Gamma) = \log\det\left(P^\intercal \, \Gamma \,P\right) $$
is always concave as a function of the the entries along the diagonal of $\Gamma$.


The cases $k = 1$ and $k = n$ are both straight-forward to check, but the in-between cases seem more difficult.



Question about the log-det function
Log det is known to be concave as a function on the positive semidefinite cone (there are many proofs, the one I like best is by way of Chandler Davis' theorem (see [this preprint](http://arxiv.org/abs/math/0208223), or Davis' original paper cited there). Since your set is a linear subspace, the affirmative answer follows, unless I am missing something in the question.


