Investigating a $1/2$-dimensional sphere

The volume of an Euclidian sphere in $n$-dimensions with radius $r$ equals $$V_n=r^n \frac{\pi^{n/2}}{\Gamma(\frac{n}{2}+1)}.$$ This formula is valid for $n\in\mathbb{N_0}$. Now I am asking myself if I can extend the definition to the rational numbers. Let's define $p=1/2$ and let $r=1$. The formula above, regardless of what it says about spheres, can be evaluated to $$V_{1/2} = 1^{1/2}\frac{\pi^{1/4}}{\Gamma(\frac{1}{4}+1)}=\frac{4\sqrt[4]\pi}{\sqrt{2 \varpi\,\sqrt{2 \pi}}},$$ where $\varpi$ is the lemniscatic constant. (It is equal to $\pi\cdot G$, $G$ is the Gauss constant). So let's assume that the volume of a $1/2$-dimensional unit sphere is equal to the expression above. Keep all of this in mind I will come back to it.

To measure dimensions of "objects" (sorry for the non-rigorosity) we can use the Hausdorff dimension. The Hausdorff dimension of a circle is two, the Hausdorff dimension of a cube is three etc., you get it. The nice thing about the Hausdorff dimension is, that it also can measure non-integer dimensions, for example the Koch snowflake that has the dimension $\frac{\log 4}{\log 3}$. Assume we can define a $1/2$-dimensional sphere with the Hausdorff dimension. With enough axioms we can then find properties of this sphere. So here is my question:

Can we find the volume of a $1/2$-dimensional sphere by using the Hausdorff dimension, and if yes, is it equal to the value $V_{1/2}$ we arrived at earlier?

Since no one has answered this question yet, let me write what I think about this interesting, but ill posed (in my opinion) question. First of all, In mathematics there are multiple meaning of dimension depending on the object we are working with. Here you are trying to incorporate two different "dimensions" (dimensional of a real vector space and Hausdorff dimension of a metric space) and see if one extend the other while making sense of volume. This may or may not possible, and most likely not. For a simple example, consider the embedding $\mathbb{N}\hookrightarrow\mathbb{Z}.$ Using $\mathbb{N}$ we count thing, like $5$ apples while $-5$ apples makes no sense anymore in the extension $\mathbb{Z}.$ Similarly the meaning of dimension turn out to be different when one looks at more general spaces.

If we merely start with a space with Hausdorff dimension $1/2,$ the volume of a sphere must means its Hausdorff_measure. Consider the generalized Cantor set obtained by removing at the middle $1/2$ segment of $[0,1],$ and in the $n$th iteration remove the central interval of length $1/2L_{n-1}$ from each remaining segment. This is an example for a this kind of space. Now you can test your conjecture $$V_{1/2} =2^{5/4}\varpi^{-1/2}$$ by an explicit computation. I will leave the rest for someone familiar with Geometric measure theory. But keep in mind that there is no canonical space with Hausdorff dimension $1/2$ that we could declare to be $\mathbb{R}^{1/2}.$ So, this is more likely to be a dead-end.

Another less promising way to approach this problem is understand what is a vector spaces of fractional dimension. Clear this will not be a natural generalization of the existing concept. For example, a linear transformation between two vector spaces is a matrix (after choosing a basis for the domain). Then, what would be a factional dimensional matrix?

Under the assumption that there is a some "fractional" analogue vector spaces: A priori, there's no reason to believe that they'd look like typical fractals as this is an algebraic approach oppose to previous geometric approach. In general, the dimension of a vector space is the cardinality of any basis. Rephrasing this definition in other ways (such as trace of the identity operator, etc.) allow us to generalize it in some cases. For example, there is already a well established notion of super vector spaces with negative integer dimensions in this modified sense (and generalizes to dualizable object in a symmetric monoidal category; see). But so far there is no sensible distinguished class of "fractional dimensional vector spaces". Even if one manage to identify them, then we need a nice Lebesgue measure to measure volumes of subsets. So, lot more things are still waiting in this road to be discovered, if possible.

Near the end of writing this answer, I found this question with two well written answers along the same line as mine. Hope it will help to clear the scenario more.