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I have a question about the convexity of an Wasserstein ambiguity set.

Let $W_1(\mu, \nu)$ be Wasserstein distance of order $1$ between $\mu$ and $\nu$ defined as $$W_1(\mu, \nu) := \min\limits_{\gamma \in \Gamma(\mu, \nu)} \bigg \{ \int_{\Xi \times \Xi} d(\xi, \zeta) \gamma(d\xi, d\zeta) \bigg \} $$ where $\Gamma(\mu, \nu)$ denote a set of all probability measures on $\Xi \times \Xi$ with marginals $\mu$ and $\nu$.

Let $\nu$ be the empirical distribution. The Wasserstein ambiguity set $\mathcal{M}$ is defined by $$\mathcal{M} := \{ \mu \in \mathcal{P}(\Xi) : W_1(\mu, \nu) \leq \theta \}.$$ where $\theta$ is given radius.

I am curious about the set $\mathcal{M}$ is convex. I notice that Wasserstein distance satisfies the triangle inequality, but I'm not sure that the set $\mathcal{M}$ is convex.

Is the Wasserstein ambiguity set of order $1$ is convex?

I have a question about the convexity of an Wasserstein ambiguity set.

Let $W_1(\mu, \nu)$ be the Wasserstein distance of order 1 between $\mu$ and $\nu$, defined as $$W_1(\mu, \nu) := \min\limits_{\gamma \in \Gamma(\mu, \nu)} \bigg \{ \int_{\Xi \times \Xi} d(\xi, \zeta) \gamma(d\xi, d\zeta) \bigg \}, $$ where $\Gamma(\mu, \nu)$ denotes the set of all probability measures on $\Xi \times \Xi$ with marginals $\mu$ and $\nu$.

Let $\nu$ be the empirical distribution. The Wasserstein ambiguity set $\mathcal{M}$ is defined by $$\mathcal{M} := \{ \mu \in \mathcal{P}(\Xi) : W_1(\mu, \nu) \leq \theta \},$$ where $\theta$ is a given radius.

I am curious about whether the set $\mathcal{M}$ is convex. I notice that Wasserstein distance satisfies the triangle inequality, but I'm not sure that the set $\mathcal{M}$ is convex.

Is the Wasserstein ambiguity set of order 1 always convex?

I have a question about the convexity of an Wasserstein ambiguity set.

Let $W_1(\mu, \nu)$ be Wasserstein distance of order 1 between $\mu$ and $\nu$ defined as $$W_1(\mu, \nu) := \min\limits_{\gamma \in \Gamma(\mu, \nu)} \bigg \{ \int_{\Xi \times \Xi} d^p(\xi, \zeta) \gamma(d\xi, d\zeta) \bigg \} $$ where $\Gamma(\mu, \nu)$ denote a set of all probability measures on $\Xi \times \Xi$ with marginals $\mu$ and $\nu$.

Let $\nu$ be the empirical distribution. The Wasserstein ambiguity set $\mathcal{M}$ is defined by $$\mathcal{M} := \{ \mu \in \mathcal{P}(\Xi) : W_1(\mu, \nu) \leq \theta \}.$$ where $\theta$ is given radius.

I am curious about the set $\mathcal{M}$ is convex. I notice that Wasserstein distance satisfies the triangle inequality, but I'm not sure that the set $\mathcal{M}$ is convex.

Is the Wasserstein ambiguity set of order 1 is convex?

I have a question about the convexity of an Wasserstein ambiguity set.

Let $W_1(\mu, \nu)$ be Wasserstein distance of order $1$ between $\mu$ and $\nu$ defined as $$W_1(\mu, \nu) := \min\limits_{\gamma \in \Gamma(\mu, \nu)} \bigg \{ \int_{\Xi \times \Xi} d(\xi, \zeta) \gamma(d\xi, d\zeta) \bigg \} $$ where $\Gamma(\mu, \nu)$ denote a set of all probability measures on $\Xi \times \Xi$ with marginals $\mu$ and $\nu$.

Let $\nu$ be the empirical distribution. The Wasserstein ambiguity set $\mathcal{M}$ is defined by $$\mathcal{M} := \{ \mu \in \mathcal{P}(\Xi) : W_1(\mu, \nu) \leq \theta \}.$$ where $\theta$ is given radius.

I am curious about the set $\mathcal{M}$ is convex. I notice that Wasserstein distance satisfies the triangle inequality, but I'm not sure that the set $\mathcal{M}$ is convex.

Is the Wasserstein ambiguity set of order $1$ is convex?