An interesting "closure" property of convex sets in **R**^{d}. 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 S_{1},...,S_{n} be a finite family of nonempty convex sets in **R**^{d}, n≥d+1. If *every* d+1 sets S_{i1},...,S_{id+1} intersect then all n sets intersect.