# Projection of closed convex sets

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

• What projection are you considering specifically? If I'm not mistaken it is also a necessary condition that the union must be convex, if you want the property to hold for any projection – Del Oct 5 '16 at 9:07
• Thank you. The answer to this question ended up to be a publication. See link.springer.com/article/10.1007/s11228-019-00525-0 – Alexey Kushnir Mar 11 at 18:50

## 1 Answer

Let $$C_1$$ and $$C_2$$ be two closed convex sets such that $$C_1\cap C_2\neq\varnothing$$, and let $$H$$ be a vector subspace with projector $$P_H$$. It always holds that $$P_H(C_1\cap C_2)\subset P_H(C_1)\cap P_H(C_2).$$ Proof: \begin{aligned}P_H(C_1\cap C_2)&=\{\textrm{Argmin}_{x\in H}\|x-y\|\, | \, y\in C_1\cap C_2\}\\ &\subset \{\textrm{Argmin}_{x\in H}\|x-y\|\, | \, y\in C_1\}\cap \{\textrm{Argmin}_{x\in H}\|x-y\|\, | \, y\in C_2\}. \end{aligned}

However, the converse is not always true. In this graph, $$H$$ is the horizontal axis while $$C_1$$ and $$C_2$$ are two line segments who intersect at a single point. The projection of their intersection is a single point, while the intersection of their projections is also a line segment.

• Thank you, but I was more interested in the other direction. – Alexey Kushnir Mar 11 at 18:50
• @AlexeyKushnir Could you please clarify -- by "other direction" do you mean the reverse inclusion? As shown above, this is false in general. – Zim Mar 12 at 17:40
• Yes, the reverse inclusion. I mean the conditions on the sets and the projector operator when the reverse inclusion holds. – Alexey Kushnir Mar 12 at 19:12