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Consider the space of probability densities $(P(\mathbb{R}^d), W_2)$ (probability measures on $\mathbb{R}^d$ with 2-Wasserstein distance).

How can we determine if a subset $Q$ is convex?

I know that a set $A$ is convex iff for all $x,y \in A$, the point on the 'line segment' $\lambda x + (1-\lambda) y$ for $\lambda \in [0,1]$ is contained in $A$.

  1. Let's say $Q$ = {mixtures of $N$ Gaussians}. Do we know the optimal transport between any two mixtures of $N$ gaussians? Is the 'line segment' between them also a mixture of Gaussians along the optimal transport path?

  2. Let's say $Q$ = {product densities} (i.e. densities that have independent marginals). What about the 'line segment' in W2 space here?

I don't have much intuition about the optimal transport plan between two distributions. Any help would be appreciated, thanks!

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    $\begingroup$ I don't understand. Convexity is a purely algebraic property. Where comes the 2-Wasserstein distance in? $\endgroup$
    – Dieter Kadelka
    Commented Jun 17, 2021 at 16:46
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    $\begingroup$ The line segment between two mixtures of $N$ Gaussians consists of mixtures of $2N$ Gaussians. $\endgroup$
    – Robert Israel
    Commented Jun 17, 2021 at 16:53
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    $\begingroup$ Not quite a counterexample to assertion 2. (no densities, but this can be solved): $\mu :=\frac{1}{2} \delta_{(0,0)} + \frac{1}{2} \delta_{(1,1)}$ is a convex combination of two probability measures with independent marginals, but $\mu$ is not of this form. $\endgroup$
    – Dieter Kadelka
    Commented Jun 17, 2021 at 17:08
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    $\begingroup$ On the one hand, you can take the simple convex combination of two measures $\mu, \nu$ as $\lambda \mu + (1-\lambda) \nu$. On the other hand you can take the displacement interpolation motivated by the Wasserstein distance: Take $\pi \in \Pi(\mu, \nu)$ to be an optimizer of $W_2(\mu, \nu)$, then if $(X, Y) \sim \pi$, the interpolation is the distribution of $\lambda X + (1-\lambda) Y$. Notice that the two concepts are very different even for Dirac measures (they result in either $\lambda \delta_x + (1-\lambda) \delta_y$ or $\delta_{\lambda x + (1-\lambda) y}$). What do you mean? $\endgroup$
    – Steve
    Commented Jun 18, 2021 at 6:56
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    $\begingroup$ @900edges, there is a notion called either "displacement convexity" or "geodesic convexity" which I believe is what you are looking for. $\endgroup$
    – pseudocydonia
    Commented Jun 19, 2021 at 5:57

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