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Optimal free export/import regions
Published online by Cambridge University Press: 17 September 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 the 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 a 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.
MSC classification
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- © Canadian Mathematical Society 2020
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