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Recently a lot of work has been done on geometric means of positive definite matrices (see here and here for example). Has anyone extended this concept to larger sets of matrices (copositive, for ...

Consider the space $X$ of all scalar products on $\mathbb{R}^n$. For a scalar product $s$ and a base $B:=b_1\ldots,b_n$ let $M_{s,B}$ denote the matrix, whose $(i,j)$-th entry is $(s(b_i,b_j))$ . ...