Introduction:

Recently, I wondered whether we could define hypercubes with non-integer dimension. It occurred to me that this would require a generalisation of the usual Cartesian Product to fractional dimensions.

A few Google searches indicated that previous work [1], [2] has been done on this subject by Ron C. Blei. However, I usually try to develop my own ideas first as this sometimes allows me to develop a perspective that is particularly insightful. For this problem I decided to start by considering hypercube volumes.

Hypercube volumes:

If the volume of a regular hypercube with integer dimension is given by:

$$\forall n \in \mathbb{N}, \text{Vol}([-1,1]^n) = 2^n$$

then I think we may define the volume of hypercubes with non-integer dimension as follows:

$$\forall x \in \mathbb{R}_+ \setminus \mathbb{N}, \text{Vol}([-1,1]^x) = 2^x$$

but the challenge is how should we define $[-1,1]^x$ analytically so that this hypercube reduces to the usual hypercube when $x \in \mathbb{N}$. I think this requires a suitable representation of the Cartesian Product.

One idea that occurred to me was to represent Cartesian Products as multipartite graphs.

References:

1. Ron C Blei. Fractional cartesian products of sets. 1979.
2. Ron Blei, Fuchang Gao. Combinatorial dimension in fractional Cartesian products. 2005.