Log concavity for matrix-valued functions and a matrix-valued Pr\'ekopa theorem

We give two different definitions of what it means for a matrix-valued function to be log concave, guided by similar notions in complex differential geometry. After discussing a few simple examples, we proceed to develop some of the basic properties associated with these new concepts. Finally, we prove a matrix-valued Pr\'ekopa theorem using a weighted, vector-valued Paley-Wiener theorem, and positivity properties of direct image bundles.