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We show that every isoperimetric set inℝ  N    {\mathbb{R}^{N}}    with density is bounded if the density is continuous and bounded above and below. This improves the previously known boundedness results, which basically needed a Lipschitz assumption; on the other hand, the present assumption is sharp, as we show with an explicit example. To obtain our result, we observe that the main tool which is often used, namely a classical “ε-   -    ε” property already discussed by Allard, Almgren and Bombieri, admits a weaker counterpart which is still sufficient for the boundedness, namely, an “ε-   -ε  β    \varepsilon^{\beta}   ” version of the property. And in turn, while for the validity of the first property the Lipschitz assumption is essential, for the latter the continuity alone is enough. We conclude by deriving some consequences of our result for the existence and almost-everywhere regularity of isoperimetric sets.

The ε - ε β property, the boundedness of isoperimetric sets in ℝ N with density, and some applications