An interesting "closure" property of convex sets in Rd. It's similar to a characterisation of compact sets (but does not follow from or imply it!). See Caratheodory's theorem and Radon's theorem (on convex sets; other theorems are also named for them!) for related theorems which relate convex sets to their dimension.
We say that a family of sets intersects if their intersection is nonempty.
Theorem. Let S1,...,Sn be a finite family of nonempty convex sets in Rd, n≥d+1. If every d+1 sets Si1,...,Sid+1 intersect then all n sets intersect.