# Does 1-Wasserstein metric have bounds?

#### jer

##### New Member
Given $$l_1(u,v)=\int_{-\infty}^{+\infty}|U-V|$$ the 1-Wasserstein distance with $$u$$ and $$v$$ two probability distributions and $$U$$ and $$V$$ their respective CDFs. The values of $$u$$ and $$v$$ are all positive in my case.

Let's assume that I have already computed $$a=max(u)$$, $$b=max(v)$$, $$c=min(u)$$, and $$d=min(v)$$,
Does $$l_1$$have an upper and a lower bound?

#### Dason

Definitely has a lower bound of 0

#### jer

##### New Member
Yes it's true, what's about the upper bound?

#### fed2

##### Active Member
Yes it's true, what's about the upper bound?
Im going 'no' since the U,V values at mins or maxes could be as far apart as you like.

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