Not an answer, just a thought.

It seems a bit too much to hope for.
Here are 300 points randomly sprinkled on the surface of an ellipsoid
in $\mathbb{R}^3$, and then projected to
two dimensions. No matter how the projection plane is oriented, the projected points are not
near the boundary of a two-dimensional convex set, but rather fill in the interior of such a set.

<br />&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;![Ellipsoid][1]


  [1]: https://i.sstatic.net/vgPQj.jpg