MathDB

Given finitely many open half planes on the Euclidean plane. The boundary lines of these half planes divide the plane into convex domains. Find a polynomial

C(q)

of degree two so that the following holds: for any

q\ge 1

integer, if the half planes cover each point of the plane at least

q

times, then the set of points covered exactly

q

times is the union of at most

C(q)

domains.
(translated by L. Erdős)

Problem Statement