Extensions of Brunn-Minkowski's inequality to multiple matrices

In 2007, Yuan and Leng gave a generalization of Ky Fan's determinantal inequality, which is a refinement of the fundamental Brunn-Minkowski inequality (det⁡(A+B))1/n≥(det⁡A)1/n+(det⁡B)1/n, where A and B are positive semidefinite matrices. In this paper, we first give an extension of Yuan-Leng's result to multiple positive definite matrices, and we further extend the result to a larger class of matrices whose numerical ranges are contained in a sector. Our result improves a recent result of Liu (2016) [16]. © 2020 Elsevier Inc.