Covering Polygonal Annuli by Strips

In 2000 A. Bezdek asked which plane convex bodies have the property that whenever an annulus, consisting of the body less a sufficiently small scaled copy of itself, is covered by strips, the sum of the widths of the strips must still be at least the minimal width of the body. We characterise the polygons for which this is so. In this note we will give a complete answer for convex polygons to the following question of A. Bezdek. Question (A. Bezdek [2, 3]). Given a plane convex body C, is there an ε > 0 such that whenever an annulus resulting from the removal of an ε-scaled copy of C from the interior of C is covered by finitely many strips, the sum of the widths of those strips must be at least the minimal width of C? Here and throughout, strips are closed, and minimal width means the shortest distance between two different parallel supporting lines. It is an elementary fact that there is at least one chord of C joining, and perpendicular to, these support lines. In 1951 T. Bang [1] solved the Tarski plank problem, showing that to cover the whole of a convex region one needs strips of total width at least its minimal width. Thus the question is asking whether making a small hole in the set leaves this unchanged. It is tempting to believe that an equivalent question can be formulated by asking about removal of small discs rather than scaled copies of C; this is however not quite true, since a C-shaped hole may be cut so close to the boundary that the disc circumscribing it protrudes outside C. Also note that if the question were asked not about small diameter but small area, then the answer would clearly be no: we can always cut a hole of arbitrarily small area that does reduce the total width of strips needed, simply by making it long and thin, perpendicular to a minimal-width chord, and with both ends very near to the boundary. Bezdek himself answered the question in the affirmative for a special class of polygons. Theorem 1 (A. Bezdek [2]). Let P be a convex polygon whose inradius is exactly half of its minimal width. Then there is an ε > 0 such that whenever any annulus resulting from the removal of an ε-scaled copy of P from the interior of P is covered by finitely many strips, the sum of the widths of those strips must be at least the minimal width of P . So in particular, the answer to the question is yes for regular polygons with an even number of sides. Zhang and Ding have announced a positive result for parallelograms, giving an explicit bound for ε in terms of the geometry of the parallelogram, [5]. Our theorem, which characterises those polygons having the property, is the following. Theorem 2. Let P be a convex polygon. (i) If there is a minimal-width chord of P that meets a vertex of P and divides the angle at that vertex into two acute angles, then for every ε > 0 an ε-scaled copy of P can be removed so that the resulting annulus can be covered by strips of total width strictly less than the minimal width of P . Supported by a Royal Society Dorothy Hodgkin Fellowship