# 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: $\alpha W_1(\mu,\lambda_\alpha) + (1-\alpha)W_1(\nu,\lambda_\alpha) \le W_1(\mu,\nu)$.

• The McCann interpolation satisfies the inequalities as equalities for $p$-Wasserstein distance with $p>1$. See Section 7.2 of the book Gradient flows by Ambrosio, Gigli and Savare. – O. Richard Aug 17 '18 at 20:34
• Indeed. Thanks! If you post your comment as an answer, I'll be happy to up-vote and accept it. – dohmatob Aug 17 '18 at 20:58

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 $W_p(\mu,\lambda_\alpha) + W_p(\lambda_\alpha,\nu)=W_p(\mu,\nu)$.