The typescript of these notes is dated 28 Nov 1985. The lemma may be useful in computer simulations of leyhunting. MB, May 2013
Let K_{1},…,K_{r} be bounded closed convex plane sets, none of which is contained in the intersection of the rest. Suppose there exists a straight line L such that

(1) 
Then we may assume that L is an external tangent to two of the K_{i}.
Proof Suppose not. Regard L as a movable line. In L, fix an origin O and a direction Ox. Let the words “left” and “right” refer to an observer looking along Ox. Let L∩K_{i} when nonempty be the interval p_{i}≤x≤q_{i}. We show that for any positive integer s there exist a labelling of the K_{i} and a position of L such that

(2)  

(3)  

(4) 
First move L to the left without rotation until (1) fails. In the limiting position L is tangent to some K_{t}. Label so that t=1; then (2) to (4) hold with s=1.
Suppose (2) to (4) hold for some s. Let K lie to the right (resp. left) of L. Roll L clockwise (resp. anticlockwise) round the boundary of K_{s} and let θ be the resulting azimuth of Ox.
If (1) holds for all θ then K_{s} is contained in every other K_{i}, contrary to hypothesis; hence (1) eventually fails. In the limiting position L is tangent to some K_{t} with t≠s. By hypothesis [i.e. since the result is assumed false] K_{t} must lie to the left (resp. right) of L. Hence L can be rolled through a further small angle to a position L′ that separates K_{s} and K_{t} with L′∩K_{t}=∅. Hence K_{s}∩K_{t}=0 and q_{s}<p_{t}.
As L moves to the limiting position, the p_{i} and q_{i} vary continuously with θ, and by (3) q_{i}≠p_{i}_{+1} throughout. Hence at the limiting position (4) holds, so that p_{i}<p_{t} for 1≤i≤s. Hence we can label so that t=s+1. This completes the inductive step.
Hence for all s there exist s distinct K_{i} : contradiction.
Remark If there are infinitely many K_{i} the result need not hold; e.g. let K_{i} be the line segment joining (i,±1/i).