The Center Lemma (Lemma 6 (page 14) from the reference paper) states that there are O(n) positions where the center of a thinnest feasible annulus with fixed inner radius covering a specified point set (with n points) may be located. Before jumping into the main assertion and proof of the Center Lemma, it is useful to introduce several new concepts: the Farthest Point Voronoi Diagram and the Boundary of the Union of Open Discs.
For a given point set, consider the set of open discs with the defined radius (the radius specified for the inner circle) that are centered on the points in the point set. Clearly the center of the minimum width annulus can not be located in any of these discs. (If it was then it would not cover at least one point.) Thus the center of the minimum width annulus must be located outside the union of all these disks. In this lemma we will show that the center of the minimum width annulus may be located on the boundary of the complement of the union of open discs (with the specified radius and centered on the points in the point set).
If there are three points on the outer circle then the center must be a vertex of the Farthest Point Voronoi Diagram (satisfying item 2 of the lemma). Figure 1 shows an annulus that is defined by three points on its outer circle. The figure also shows its Farthest Point Voronoi Diagram (in green).
On the other hand if there is a diametral pair of points on the outer circle then the pair of points must satisfy item 1 of the lemma. Figure 2 shows an annulus that is defined by a diametral pair. The line segment between the diametral pair is shown in green. Clearly this line segment represents the diameter of the circle as well as the point set.
Clearly the center satisfies item 4. Figure 3 shows an annulus that is defined by two points on the inner circle and one on the outer circle. The two open discs around the two inner circle points are shown in green. It should be clear that the common center must be on the boundary of the green discs.
Since the common center is equidistant from the two points on the outer circle, it must lie on a Farthest Point Voronoi Edge. Since there is at least one point on the outer circle, the common center lies on the Boundary of the Complement of the Union of Open Discs. Figure 4 shows an annulus that is defined by two points on the outer circle and one on the inner circle. The blue arrow shows one Farthest Point Voronoi Edge and the green disc is one of the Open Discs.