I have been thinking about a couple different problems in fractal geometry (including I one deleted because it was ill posed) and realize they all depend in a fundamental way on the problem of: Can we construct a fractional dimensional sphere?
I did some digging online and kept coming up short, there is a post here asking about spheres of fractal dimension but unfortunately no explicit discussions about construction take place.
So I wanted to pose the problem here explicitly. We begin with a little bit of setup:
We can define a "n-ball" in 3 different ways I suspect when we consider fractals these three ways will end up becoming inequivalent.
Strategy 1: Points that are equidistant from a center
We can always define a ball of radius $\epsilon$ on a set $S$ equipped with a distance function $d$ as the set of all points of distance $\epsilon$ away from some center point $u \in S$. This certainly works and lets you make balls in very abstract settings. But we run into some challenges with this. The "Usual" spheres (the sphere, circle, line segment, etc... ) are defined as balls in $\mathbb{R}^n$ so to create a canonical sphere we need to be able to generalize $\mathbb{R}^n$ for fractional $n$ and to define that (to the best of my ability) we should know how to construct fractional-dimensional cubes (but to define these cubes we need angles and to define angles we need spheres again). If only there was a way to define fractal $\mathbb{R}^n$ that's natural and doesn't depend on spheres we could then define our spheres on it.
Strategy 2: Spheres as fractal manifolds
We can define an $\alpha > 1$ sphere $S$ as a set with a few properties.
- $S$ has a boundary $\partial S$ is not empty and $\partial^2 S = \emptyset$
- $S$ has hausdorff dimension $\alpha$ and moreover its hausdorff measure should be equal $V_s(d) = \frac{\pi^{\frac{n}{2}}}{\Gamma \left( \frac{n}{2} + 1 \right)} \frac{d^n}{2^n}$ (I'm defining my hausdorff content scaled so that it agrees with lebesgue measure) where $d$ is the diameter of the set (the length of the largest $1$-dimensional line segment we can fit in the set)
- $S$ is simply connected (I think for $\alpha<1$ this requirement must be dropped)
- The boundary of $S$ has hausdorff dimension $\alpha-1$ and its hausdorff measure is $2 \frac{\partial}{\partial d} V_s(d)$.
These are very specific requirements and they don't make it obvious what a "construction" looks like for fractional dimensions. And/Or i'm just incompetent with using these tools.
Strategy 3: Spheres Maximize Their Volume For a Given Boundary
This is again non constructive (unless we can define a calculus of variations in fractional dimensional space) but the premise is to start with a set $\partial S$ which has a fixed fractal dimension and ask what set should $\partial S$ be so that it is the boundary of a set $S$ and the hausdorff measure of $S$ is maximized.
The Question: Using ANY of these strategies (or your own if you have a better characterization that is easier to generalize) have we been able to define a family of fractional $n$-spheres? Or even just been able to construct ONE such sphere for non positive-integral dimension?
It appears that cantor sets have a known hausdorff measure of 1 (I need to see which definition is used but this might re-scale to the desired spherical measures)
