Asymptotics of the $s$-perimeter as $s\searrow 0$

We deal with the asymptotic behavior of the $s$-perimeter of a set $E$ inside a domain $Ω$ as $s\searrow0$. We prove necessary and sufficient conditions for the existence of such limit, by also providing an explicit formulation in terms of the Lebesgue measure of $E$ and $Ω$. Moreover, we construct examples of sets for which the limit does not exist.

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