Extensions of Brunn-Minkovski's inequality to multiple matrices

Abstract

Yuan and Leng (2007) gave a generalization of Ky Fan's determinantal inequality, which is a celebrated refinement of the fundamental Brunn-Minkowski inequality $(\det (A+B){)}^{1/n}\ge (\det A{)}^{1/n}+(\det B{)}^{1/n}$, where $A$ and $B$ are positive semidefinite matrices. In this note, we first give an extension of Yuan-Leng's result to multiple positive definite matrices, and then 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 [Linear Algebra Appl. 508 (2016) 206--213].

arXiv comment10 pages. This is the final version. Any suggestions and comments are welcome. E-mail addresses: [email protected] (Yongtao Li)