*Accepted Paper*

**Inserted:** 25 oct 2020

**Last Updated:** 25 oct 2020

**Journal:** Canadian Mathematical Bulletin

**Year:** 2020

**Abstract:**

We consider the problem of finding two free export$/$import sets $E^+$ and $E^-$ that minimize the total cost of some export$/$import transportation problem (with export$/$import taxes $g^\pm$), between two densities $f^+$ and $f^-$, plus penalization terms on $E^+$ and $E^-$. First, we prove existence of such optimal sets under some assumptions on $f^\pm$ and $g^\pm$. Then, we study some properties of these sets such as convexity and regularity. In particular, we show that the optimal free export (resp. import) region $E^+$ (resp. $E^-$) has boundary of class $C^2$ as soon as $f^+$ (resp. $f^-$) is continuous and $\partial E^+$ (resp. $\partial E^-$) is $C^{2,1}$ provided that $f^+$ (resp. $f^-$) is Lipschitz.

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