Let X be a separable Banach space. The p-Wasserstein space on X is defined as...

Madison Costa

Madison Costa

Answered

2022-11-21

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 ) < } , p 1

Answer & Explanation

Antwan Wiley

Antwan Wiley

Expert

2022-11-22Added 13 answers

Step 1
For X = R d , you can choose the Euclidean norm or the 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 ) <  for each  1 i d }
Step 2
where μ i is the i-marginal: μ i ( B ) = μ ( R i 1 × B × R d i )

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