Suppose x1 x2 xn sub r2 are convex sets such that

Question

Math 121c: Topics in Geometric Combinatorics, Spring 2012 Problems-

(a) Suppose X₁, X₂, . . . , X_n ⊂ R² are convex sets such that through every four of them there exists a line intersecting them. Prove that there exist two lines l₁ and l₂ such that every X_i intersects l₁ or l₂.

(b) Let X₁, . . . , X_n ⊂ R^d be full dimensional rectangular prisms with faces parallel to the coordinate axes. Determine, which proof, a function f(d), dependent only on d, such that the following statement is true:

"Suppose that for any f(d) of these convex sets, there exists a hyperplane intersecting them. Then there exists a hyperplane intersecting all X_i, 1 ≤ i ≤ n.", or prove such a function cannot exist.