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We introduce a new family of fractal dimensions by restricting the set ofdiameters in the coverings in the usual definition of the Hausdorff dimension.Among others, we prove that this family contains continuum many distinctdimensions, and they share most of the properties of the Hausdorff dimension,which answers negatively a question of Fraser. On the other hand, we also provethat among these new dimensions only the Hausdorff dimension behaves nicelywith respect to H\"older functions, which supports a conjecture posed by Banajiobtained as a natural modification of Fraser's question.  We also consider the supremum of these new dimensions, which turns out to bean other interesting notion of fractal dimension.  We prove that among those bilipschitz invariant, monotone dimensions on thecompact subsets of $\mathbb{R}^n$ that agree with the similarity dimension forthe simplest self-similar sets, the modified lower dimension is the smallestand when $n=1$ the Assouad dimension is the greatest, and this latter statementis false for $n>1$. This answers a question of Rutar.

New Hausdorff type dimensions and optimal bounds for bilipschitz invariant dimensions