On The Linearization of Alternative Means

Alternative means have recently attracted considerable attention in matrix analysis and operator theory. In this paper, we investigate the linearization problem for alternative means, namely the question of determining when a mean can be expressed as an affine combination of the matrices under consideration. We first prove a conjecture of Choi, Kim, and Lim for the Wasserstein mean. More precisely, we show that the Wasserstein mean $\text{A}\diamond \text{B}$ is linearizable if and only if $\text{AB}=\text{BA}$ and $\mid \text{Spec}({\text{A}}^{-1}\text{B})\mid \le 2$. We further establish a general rigidity theorem for a large class of alternative means. Specifically, we prove that an analogous characterization holds whenever the representing function $f$ is of the form $f(x)=\sqrt{h(x)}$, where $h$ is a non-affine operator monotone function. As consequences, we obtain linearization criteria for several families of alternative means, including logarithmic, harmonic, and power-type means.