In 1957, Hadwiger proved that if an ordered finite family of pair wise disjoint convex sets in the plane has the property that for every 3 sets, there is a line intersecting them in their relative order, then there is a line intersecting all the sets in the family. This was later generalized to the case where we wish to intersect a family of convex sets in d dimensions with a hyperplane by Goodman, Pollack, and Wenger in a series of 3 papers.

We provide an answer to a longstanding open problem of determining a necessary and sufficient condition for a family of convex sets in d dimensions that determines the existence of a k-dimensional affine subspace that intersects each set in the family. This generalizes the previously mentioned results.

This is joint work with Nikola Sadovek.