Every explanation I read so far is quite incomprehensible. I’m especially interested in its relation to the Hausdorff-dimension and fractals.
Thanks in advance.
In: Mathematics
This is one of several ways of extending the notion of “dimension” to a broader class of mathematical spaces than it would usually apply to. As the name suggests, the definition is based on coverings of the object under consideration. For example, [consider this circle](https://upload.wikimedia.org/wikipedia/commons/1/16/Refinement_on_a_circle.png). On the right we see a covering of this circle using four sets. At any given point, as many as three of these sets overlap. However, if we shrink down each of the four sets involved, we can get the covering on the left, where only as many as two of the sets overlap at any given point. In fact it turns out that no matter what covering you start out with for the circle, you can *always* shrink the sets down so that only two overlap at any given point, while still preserving the fact that every point on the circle lies inside (at least) one of the sets.
On the other hand, [consider this solid box](https://upload.wikimedia.org/wikipedia/commons/b/bc/Refinement_on_a_planar_shape.png). At the top we have a similar covering with four sets, but in the center all four overlap. We see on the bottom left that we can reduce the amount of overlap to three sets at any given point, but on the bottom right we see how any attempt to reduce the overlap to only two sets results in some regions not being covered. In fact, with a solid box we can only guarantee that no more than *three* sets will overlap at any given point
This value, the guaranteed number of overlapping sets, is how we get the Lebesgue covering dimension. To make it compatible with other notions of dimension, we actually subtract 1 from it: the circle is 2-1 = 1-dimensional, the solid box is 3-1 = 2-dimensional, etc. For normal spaces and shapes, the Lebesgue covering dimension corresponds exactly with all the other definitions of “dimension”. Lines and curves are 1-dimensional; solid polygons, the surface of a sphere, and an infinite plane are all 2-dimensional; solid polyhedrons are 3-dimensional; and so on. For some weird fractal shapes, the Lebesgue covering dimension is distinct from other concepts like the Hausdorff dimension. For one thing, the Lebesgue covering dimension must by definition be an integer, while the Hausdorff dimension can be any positive real number. In these cases the Hausdorff dimension is usually larger than the Lebesgue covering dimension. The Lebesgue covering dimension is also a more general definition, in that there are some spaces where it’s impossible to talk about the Hausdorff dimension (for example, if there’s no well-defined notion of “distance” in the space) but the Lebesgue covering dimension still works as a definition because it relies only on this notion of covering a space with sets.
Latest Answers