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We prove the validity of theε  -   ε  β{\varepsilon-\varepsilon^{\beta}}    property in the isoperimetric problem with double density, generalising the known properties for the case of single density. As a consequence, we derive regularity for isoperimetric sets.

The ε - ε β Property in the Isoperimetric Problem with Double Density, and the Regularity of Isoperimetric Sets