Question 1
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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.** 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 ?


Question 2
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In case the answer is negative, are there any other metrics (or just divergences like KL, etc.) between probability measures with such a property ?


Question 3
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Same questions for the weaker condition: $\alpha W\_1(\mu,\lambda\_\alpha) + (1-\alpha)W\_1(\nu,\lambda\_\alpha) \le W\_1(\mu,\nu)$.



Wasserstein interpolation between two probability measures on a metric space
The following discussion is based on the book Gradient Flows by Ambrosio, Gigli, and Savare (2008).


Consider $p$-Wasserstein distance with $p>1$ on a Hilbert space (for the sake of uniqueness). Let $\gamma$ be the optimal transport plan between $\mu$ and $\nu$ under the $p$-Wasserstein distance. Denote by $\pi^i$ be the $i$-th projection ($i=1,2$) and by $\#$ the pushforward of measures (see Section 5.2). Define $\lambda\_\alpha = ((1-\alpha) \pi^1 + \alpha \pi^2)\_\# \gamma$. Then according to Section 7.2, we have $W\_p(\mu,\lambda\_\alpha)=\alpha W\_p(\mu,\nu)$, $W\_p(\lambda\_\alpha,\nu)=(1-\alpha)W\_p(\mu,\nu)$ and $W\_p(\mu,\lambda\_\alpha) + W\_p(\lambda\_\alpha,\nu)=W\_p(\mu,\nu)$.


