Unbounded regions in an arrangement of lines in the plane (Q1875907)

Consider an arrangement of \(n\) lines in the plane. The arrangement divides the plane into regions, the faces of the arrangement. \(2n\) of the faces of the arrangement are unbounded, but if we move the lines, unbounded faces might become bounded and bounded faces might become unbounbed. The author shows that any set of points \(P\) in the plane, no two of them in the same face, with the property that for each point \(p\in P\) there is a homotopy of the lines that avoids all points of \(P\) and makes the face that contains \(p\) unbounded, has cardinality at most \(2n\). The question is motivated by a problem in differential geometry.