Geometric matrix midranges

We define geometric matrix midranges for positive definite Hermitian matrices and study the midrange problem from a number of perspectives. Special attention is given to the midrange of two positive definite matrices before considering the extension of the problem to more than two matrices. We compare matrix midrange statistics with the scalar and vector midrange problem and note the special significance of the matrix problem from a computational standpoint. We also study various aspects of geometric matrix midrange statistics from the viewpoint of linear algebra, differential geometry, and convex optimization. A solution to the N-point problem is offered via convex optimization.