Assumptions about the convex hull of a closed path in a 4D space

[Arne]

TL;DR Summary

I have a closed path in 4d space, and I would like to prove some assumption about it's convex hull.

Hello everyone,

I am struggling to get insight into a certain set in 4D space. Given is a closed path in 4D-space with constant Euclidean norm
$$\vec{\gamma} (\theta):[0,2\pi]\to\mathbb{R}^4, \ \ \vec{\gamma}(0)=\vec{\gamma}(2\pi), \ \ ||\vec{\gamma}(\theta)||_2 = \mathrm{const.}$$
I am looking at a set which is defined as the convex hull of these points
$$S = \left\{\sum_k \alpha_k \vec{\gamma}(\theta_k): \ \ 0\leq\alpha_k\leq 1, \ \ \sum_k \alpha_k = 1, \ \ \theta_k \in [0,2\pi]\right\}$$
I'm also assuming, that the path is injective ($\vec{\gamma} (\theta_1)\neq\vec{\gamma} (\theta_2)$ for $\theta_1 \neq \theta_2$), and it does NOT lie in a 2- or 3-dimensional hyperplane; the set thus forms a 4-dimensional volume. Due to the constant norm, the path lies on the 3-dimensional boundary/surface of the volume, but since there are no straight sections in the path, it should also form the extremal points of the convex hull.

Now I have some assumptions about this set, that I can support with heuristic test. However, I would like to make sure, that these things are actually correct (I don't necessarily need to do a formal prove, but at least some sufficient arguments, why I can regard these assumptions to be true with some certainty).
1. It seems, that three terms $\sum_{k=1}^3 \alpha_k \vec{\gamma}(\theta_k)$ are enough to span the whole volume. This would give 5 independent parameters, which can be enough to parameterize a 4d volume. But it could still be, that 4 terms might be able to reach points, that are not possible with 3 terms.
2. It seems, that the 3-dimensional surface of the 4-dimensional volume are exactly the points that can be reached with at most 2 terms $\alpha_1 \vec{\gamma}(\theta_1)+\alpha_2 \vec{\gamma}(\theta_2)$. Two terms would give 3 independent parameters, that can span a 3-dimensional surface, but I don't know, why this would be the surface of the 4d-volume.
3. It seems, that the points on the surface cannot be written with 3 distinct terms (which means for 3 terms, either a $\alpha$ vanishes, or two of the angles $\theta$ are the same).

Of course, I am not expecting any complete proves. But after a frustrating lengthy google search, I would be happy over any fresh ideas or pointers for where to look further. Thanks!Arne

 

[andrewkirk]

I suggest first trying to prove these hypotheses for the simpler case of lower dimension. For hypothesis 1, try to prove that all points on the convex hull of a simple, closed, non-planar curve on the surface of an ordinary sphere can be expressed as a linear combination of two points on the curve. Following your intuition above, two points have three independent parameters, which can parametrise a 3D space. But can it parametrise that particular 3D space, ie the convex hull?
If you're able to prove that, it may give some intuitions on how to approach the 4D case.