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.

              Ellipsoid http://cs.smith.edu/%7Eorourke/MathOverflow/Ell300.jpg

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.

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

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. 

              Ellipsoid http://cs.smith.edu/%7Eorourke/MathOverflow/Ell300.jpg