If I have $n$ 2-dimensional convex "regions" that are the projections along $n$ (independent) dimensions of a n-dimensional compact subspace:
- Does that imply that the latter is convex? Is the orthogonality of the dimensions needed?
- What about the reciprocal? (I'm convinced it's true, but if you have a conter-example, shoot :)
Consider the closed unit ball, and remove from it all points a distance at most $1/10$ from the open line segment between $(0,0,0)$ and $(1,1,1)$. This set is compact and simply-connected, but all its projections are convex, despite the set not being convex.
If a set is convex, then all its projections are convex. This follows from the definition of convexity. If $x,y$ are two points in the set, then all points on the line segment between $x$ and $y$ are also in the set. When we project the set onto some plane, then the line segment is projected onto a new straight line segment from the projection of $x$ to the projection of $y$. Thus the projection of the set is convex.