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

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