Discover published sets by community

For a given dimension $s$, the $s$-dimensional Hausdorff measure is the measure obtained by the Carathéodory's Construction that corresponds to the $s$-dimensional volume of the set.

A principle stating that in a metric space, every compact subset is closed and bounded. This principle is foundational in topology and related to the properties of Hausdorff spaces.

The Hausdorff dimension is a metric that measures the local size of a space, taking into account the irregularities in its shape. It is defined as the infimum of the set of dimensions for which the Hausdorff measure of the space is zero.

A distance function defined between two subsets of a metric space, it is the greatest distance one must travel from a point in one set to reach the other set, and vice versa.

A lemma that provides conditions under which a set has positive measure, indicating that there exists a probability measure with certain decay properties on balls in a metric space.

A method for defining a measure, which can be used to define the Hausdorff measure. It involves covering a set with basic 'balls' whose diameters tend to zero and considering the infimum of the sum of a power of the diameters of these balls.

A foundational result in fractal geometry which establishes that for any subset of a metric space, there exists a unique critical dimension at which the Hausdorff measure transitions from infinity to zero.

An alternate name for the Hausdorff dimension, recognizing the contributions of mathematician Abram Samoilovitch Besicovitch in the field of geometric measure theory.

Packing measure is an alternative to Hausdorff measure that often behaves better on sets with a lot of overlapping structure, defined in a similar way to Hausdorff measure but using disjoint balls for the covering.