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Not an answer, just a thought.
It seems a bit too much to hope for. Here are 100 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.
        Ellipsoid http://cs.smith.edu/%7Eorourke/MathOverflow/Ell100.jpg