In this paper, we first prove that any interior point of an open interval of the real line can be interpreted as Fréchet means with respect to corresponding metric distances, thus extending the result of [Dinh et al., Mathematical Intelligencer 47.2 (2025)] which was restricted to intervals on the positive reals by using the family of power means: Our generic construction relies on the concept of scales of means that we demonstrate with the scale of exponential means and the scale of radical means. Second, we interpret those Fréchet means geometrically as the center of mass of any two distinct points on the Euclidean line expressed in various coordinate systems: Namely, by interpreting the Euclidean line as a 1D Hessian Riemannian manifold, we introduce pairs of dual Fréchet/Karcher means related by convex duality in dual coordinate systems. This result yields us to consider squared Hessian metrics in arbitrary dimension: We prove that these squared Hessian metrics amount to Euclidean geometry with the Riemannian center of mass expressed in primal coordinate systems as multivariate quasi-arithmetic means coinciding with left-sided Bregman centroids.

翻译：本文首先证明：实直线上任意开区间内的任意一点，均可解释为相应度量距离下的弗雷歇均值，从而推广了[Dinh等人，《数学情报员》47.2（2025）]仅通过幂平均族在正实数区间上的结果：我们的通用构造依赖于均值尺度的概念，并通过指数均值尺度与根式均值尺度加以阐释。其次，我们将这些弗雷歇均值几何地解释为欧几里得直线上任意两个相异点在多种坐标系下的质心：具体而言，通过将欧几里得直线视为一维Hessian黎曼流形，我们引入了在双坐标系中通过凸对偶关联的成对弗雷歇/卡赫均值。该结果引导我们考虑任意维度的平方Hessian度量：我们证明这些平方Hessian度量等价于欧几里得几何，其中黎曼质心在主坐标系中表现为与左偏Bregman质心重合的多元拟算术均值。