Example of an antipodal set

Question

So I am trying to solve the mentioned homework question:

Find an example of an antipodal set of size $2^d$.

The antipodal set is a finite set $F$ of points in $R^d$ such that for any two points $f_1,f_2$ there are two distinct parallel hyperplanes $H_1, H_2$ such $f_i \in H_i$ and $F \subset Conv(H_1 \cup H_2$) for example, the triangle is an antipodal set because for any two points there are two parallel hyperplanes also parallelogram

Thanks in advance.

Answer 1

If I am reading the definition correctly then the vertices of the unit cube in dimension $d$ form such a set. The two parallel hyperplanes will contain opposite faces of the cube whose dimension depends on the dimension of the smallest face containing the two points.

Imagine the possibilities in the plane and space to see the geometry.

I wonder if size $2^d$ is maximal for antipodal sets in $\mathbb{R}^d$.