I have a simply formulated geometric question that I was not able to find an answer to. Any help/references/conjectures on the topic will be highly appreciated.
Let us consider two convex closed sets in a Euclidean space. Assume they have non-empty intersection. Let us consider a projection that transforms both sets to some subspace of a smaller dimension.
When is the projection of the intersection of two closed convex sets coincides with the intersection of projections?
One trivial answer when this is the case for any possible projector operator is when union of the two sets is also convex.
Unfortunately, this answer is not sufficient for my applications (for mathematical economics). My applications consider two specific convex closed sets (one is polyhedron, the other is a cone) and a specific projection operator. I want to understand/derive some intuitive sufficient conditions on both sets and the projection operator that guarantee that the projections of the intersection equal the intersection of projections.
Any help would be appreciated.
Thank you, Alex