Let X be a separable Banach space. The p-Wasserstein space on X is defined as W p ( X ) = { μ ∈ P ( X ) : ∫ X | | x | | p d μ ( x ) &lt; ∞ } , p ≥ 1

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Let X be a separable Banach space. The p-Wasserstein space on X is defined as      W    p    (  X  )  =  {  μ  ∈  P  (  X  )  :      ∫          X            |        |    x      |              |              p        d  μ  (  x  )  &amp;lt;  ∞  }  ,  p  ≥  1

Antwan Wiley · Accepted Answer

Step 1For       X    =            R        d  , you can choose the Euclidean norm or the       ℓ    p   norm, which is equivalent, that is, there exists constants       C          p      ,      d       and       C          p      ,      d        ′   such that       C          p      ,      d                  (              ∑                  k          =          1                d            |              x        k                    |        p            )              1              /            p        ⩽            ∑              k        =        1            d              x      k      2        ⩽      C          p      ,      d        ′              (              ∑                  k          =          1                d            |              x        k                    |        p            )              1              /            p      . With this in mind, we derive that      W        (                  R            d        )    =      {    μ    ∈          P              (                        R                d            )        ,          ∫                        R                      |    x          |      p        d          μ      i        (    x    )    &amp;lt;    ∞           for each         1    ⩽    i    ⩽    d    }  Step 2where       μ    i   is the i-marginal:       μ    i    (  B  )  =  μ      (                  R                    i        −        1              ×    B    ×                  R                    d        −        i              )

Let X be a separable Banach space. The p-Wasserstein space on X is defined as W_p(X)={mu in P(X):int_X ||x||^P d mu(x)<infty}

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