# Question 1

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

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

Same questions for the weaker condition: $\alpha W_1(\mu,\lambda_\alpha) + (1-\alpha)W_1(\nu,\lambda_\alpha) \le W_1(\mu,\nu)$.