In this paper we study the monotonicity, in-betweenness and in-sphere properties of matrix means with respect to Bures–Wasserstein, Hellinger and log-determinant metrics. More precisely, we show that the matrix power means (Kubo–Ando and non-Kubo–Ando extensions) satisfy the in-betweenness property in the Hellinger metric. We also show that for two positive definite matrices A and B, the curve of weighted Heron means, the geodesic curve of the arithmetic and the geometric mean lie inside the sphere centered at the geometric mean with the radius equal to half of the log-determinant distance between A and B.

中文翻译：

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关于不同度量的矩阵均值的某些几何性质

在本文中，我们研究了相对于Bures–Wasserstein，Hellinger和对数行列式指标的矩阵均值的单调性，中间性和球内特性。更准确地说，我们证明矩阵幂均值（Kubo–Ando扩展和非Kubo–Ando扩展）满足Hellinger度量中的中间性。我们还表明，对于两个正定矩阵A和B，加权苍鹭均值曲线，算术和几何均值的测地曲线位于球体内，以几何均值为中心，半径等于对数行列式的一半A和B之间的距离。