Given probability measures $\mu$ and $\nu$ on the same metric space $X=(X,d)$, and $\alpha \in [0, 1]$, is it always possible to find another probability measure $\lambda_\alpha$ on $X$ such that $W_1(\mu,\lambda_\alpha) \le \alpha W_1(\mu,\nu)$ and $W_1(\nu,\lambda_\alpha) \le (1 - \alpha) W_1(\mu,\nu)$ ?

Observations
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Thus $\lambda_\alpha$ is some sort of interpolation between $\mu$ and $\nu$. Maybe the *Wasserstein barycenter* of $\mu$ and $\nu$ (with weights $\alpha$ and $1-\alpha$) or the *Mccann interpolation* can do the job ?