Log concavity for matrix-valued functions and a matrix-valued Prékopa theorem

Abstract

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 Prékopa theorem using a weighted, vector-valued Paley-Wiener theorem, and positivity properties of direct image bundles.

25 pages

References in corpus (1)

• Curvatures of direct image sheaves of vector bundles and applications I

Cited by in corpus (4)

• The openness conjecture and complex Brunn-Minkowski inequalities
• Curvature positivity of invariant direct images of Hermitian vector bundles
• Characterization of Curvature positivity of Riemannian metrics on flat vector bundles
• Variation of Bergman kernels of pseudoconvex domains