Subdivisions of rotationally symmetric planar convex bodies minimizing the maximum relative diameter

SUBDIVISIONS OF ROTATIONALLY SYMMETRIC PLANAR CONVEX BODIES MINIMIZING THE MAXIMUM RELATIVE DIAMETER ANTONIO CA˜ NETE, UWE SCHNELL, AND SALVADOR SEGURA GOMIS Abstract. In this work we study subdivisions of k-rotationally symmetric planar convex bodies that minimize the maximum relative diameter functional. For some particular subdivisions called k-partitions, consisting of kcurves meeting in an interior vertex, we prove that the so-called standard k-partition (given by kequiangular inradius segments) is minimizing for any k∈N,k⩾3. For general subdivisions, we show that the previous result only holds for k⩽6. We also study the optimal set for this problem, obtaining that for each k∈N,k⩾3, it consists of the intersection of the unit circle with the corresponding regular k-gon of certain area. Finally, we also discuss the problem for planar convex sets and large values of k, and conjecture the optimal k-subdivision in this case. 1. Introduction The study of the classical geometric magnitudes associated to a planar compact convex set (perimeter, area, inradius, circumradius, diameter and minimal width) is one of the main points of interest for Convex Geometry, and these magnitudes have been considered since ancient times. They appear in the origin of this mathematical field, and in fact, there is a large variety of related problems, providing a more complete understanding of these geometric measures. One of the most common problems involving these magnitudes consists of finding precise relations between some of them (and not only in the convex framework). For instance, the well-known isoperimetric inequality relates the perimeter and the area functionals, stating that, among all planar sets with fixed enclosed area, the circle is the one with the least possible perimeter [15, 14]. In the same direction, the isodiametric inequality asserts that the circle is also the planar compact convex set of prescribed area with the minimum possible diameter [2]. Another example is given by the equilateral triangle, which is the planar compact convex set with minimum enclosed area for fixed minimal width [16]. We point out that these relations are usually established by means of appropriate general inequalities, with the characterization of the equality case providing then the optimal set for the corresponding functionals Date: September 25, 2018. 2010 Mathematics Subject Classification. 52A10, 52A40. Key words and phrases. partitioning problems, k-rotationally symmetric planar convex body, maximum relative diameter. 1 2 A. CA ˜ NETE, U. SCHNELL, AND S. SEGURA GOMIS (that is, the set attaining the minimum possible value for the functional). A nice detailed description of the relations between two (and three) of these classical magnitudes can be found in [17]. Apart from the above approach, these geometric functionals can be studied in a large range of different related problems. One of these problems, which has been considered in several works during the last years, is the so-called fencing problem, or more generally, the partitioning problem. For these problems, the main interest is determining the division of the given set into kconnected subsets (of equal or unequal areas) which minimizes (or maximizes) a particular geometric magnitude [7]. The relative isoperimetric problem is one of these questions, where the goal is minimizing the perimeter of the division curves for fixed enclosed areas. For this problem, the general geometric properties satisfied by the minimizing divisions can be deduced by using a variational approach, and particular results characterizing the solutions have been obtained for some regular polygons and for the circle, see [18, 1, 4, 5]. Regarding the diameter functional, the partitioning problem can be studied by means of the maximum relative diameter functional. We recall that for a planar compact convex set Cand a decomposition of Cinto kconnected subsets C1,...,Ck, the maximum relative diameter associated to the decomposition is defined as the maximum of the diameters of the subsets Ci, for i= 1,...,k (this means that it measures the largest distance in the subsets provided by the decomposition). In this setting, the partitioning problem seeks the decomposition of Cinto ksubsets with the least possible value for the maximum relative diameter. This problem has been recently studied for k= 2 and k= 3 under some additional symmetry hypotheses [11, 3], and the results therein constitute the main motivation for our work. In [11], the previous problem is treated for centrally symmetric planar compact convex sets, and for decompositions into two subsets of equal areas, proving that the minimizing division for the maximum relative diameter is always given by a line segment passing through the center of symmetry of the set [11, Prop. 4]. We point out that a more precise characterization of such a minimizing segment is not provided. Furthermore, the optimal set for this problem (that is, the centrally symmetric planar compact convex set attaining the minimum possible value for the maximum relative diameter) is also determined up to dilations, consisting of a certain intersection of two unit circles and a strip bounded by two parallel lines, see [11, Th. 5]. In [3], the problem is considered for 3-rotationally symmetric planar compact convex sets, and for divisions into three subsets of equal areas. The main result establishes that the so-called standard trisection, determined by three equiangular inradius segments, minimizes the maximum relative diameter [3, Th. 3.5 and Prop. 5.1] for any set of that class. In addition, the optimal set for this problem is also characterized, up to dilations, as the intersection of the unit circle with a certain equilateral triangle [3, Th. 4.7]. We remark that, although the results in [3] are stated for divisions into three subsets of equal areas, all of them also hold in the case of unequal areas. MINIMIZING SUBDIVISIONS FOR THE MAXIMUM RELATIVE DIAMETER 3 This paper is inspired precisely in these last two references involving the maximum relative diameter [11, 3]. As a natural continuation, we shall focus on the class Ckof k-rotationally symmetric planar convex bodies (that is, compact convex sets) for each k∈N,k⩾3. And for each set in Ck, we shall investigate the divisions into kconnected subsets minimizing the maximum relative diameter functional. In this setting we shall consider two different types of divisions, namely k-subdivisions and k-partitions. For a given set C∈ Ck, a k-partition of Cwill be a decomposition of Cinto kconnected subsets by kcurves starting at different points of ∂C, and with all of them meeting at an interior point in C. On the other hand, a k-subdivision of Cwill be a general decomposition of Cinto kconnected subsets, with no additional restrictions. The goal of this paper is studying the minimizing k-partitions and minimizing k-subdivisions for the maximum relative diameter, for any k-rotationally symmetric planar convex body. In fact, we shall see that the so-called standard k-partition (consisting of kinradius segments symmetrically placed) is a minimizing k-partition for the maximum relative diameter, for any k∈N,k⩾3 (Theorem 4.5), being also minimizing among k-subdivisions only when k⩽6 (Theorem 4.6). For k > 7 we shall provide some examples showing that the standard k-partition is not a minimizing k-subdivision, which suggests that, in those cases, a complete characterization is a difficult question. In addition, we point out that the uniqueness of the minimizing k-partition does not hold for this problem, as explained in Subsection 4.1, since proper slight deformations of a minimizing k-partition are also minimizing. Another interesting result in this paper is the characterization of the optimal bodies for this problem. For each k∈N,k⩾3, we are able to determine the k-rotationally symmetric planar convex body attaining the minimum possible value for the maximum relative diameter, when considering k-partitions. Taking into account the previous Theorem 4.5, for each set of this class, that minimum value will be provided by the corresponding standard k-partition. In Section 5 we give the description of these optimal bodies for each k⩾3, which, in general, consist of the intersections of the unit circle with a certain regular k-gon, up to dilations. We remark that the case k= 2 (treating the decompositions into two subsets for centrally symmetric planar convex bodies and previously studied in [11]) presents some differences with respect to the other situations, as explained in Remark 3.6, and cannot be treated as for k⩾3. The main reason for this particular behaviour lies in the following technical fact: the associated standard 2-partition of a set of this class, which is provided by two inradius segments placed symmetrically, is just a line segment passing through the center of symmetry of the set, and so the computation of the maximum relative diameter associated to this standard 2-partition cannot be done by using Lemma 3.2, which is fundamental when k⩾3. We have organized this paper as follows. Section 2 contains the precise definitions and the setting of the problem, and in Section 3 we describe the standard k-partition for any k-rotationally symmetric planar convex body. 4 A. CA ˜ NETE, U. SCHNELL, AND S. SEGURA GOMIS This k-partition is constructed by applying successively the existing rotational symmetry to an inradius segment of the set. We remark that Lemma 3.2 establishes how to compute the maximum relative diameter for the standard k-partition. In Section 4 we obtain the main results of this paper. Due to Lemmata 3.2, 4.1 and 4.4, we prove our Theorem 4.5: For any k-rotationally symmetric planar convex body, the associated standard k-partition is a minimizing k-partition for the maximum relative diameter. Furthermore, regarding general k-subdivisions, we obtain Theorem 4.6: For any k-rotationally symmetric planar convex body, the associated standard k-partition is a minimizing k-subdivision for the maximum relative diameter, when k⩽6. Figure 1. The standard 3-partition for the equilateral triangle, the standard 4-partition for the square, and the standard 8-partition for the regular octagon When k⩾7, Examples 4.7 and 4.8 show that the standard k-partition is not a minimizing k-subdivision in general, suggesting that the characterization of such minimizing k-subdivisions is a difficult task which will depend on the particular considered set. We also point out that, for k= 3, the minimizing 3-partition and 3-subdivision in this setting have been previously obtained, see [3, Th. 3.5 and Prop. 5.1]. Moreover, in Subsection 4.1 we discuss the uniqueness of the solutions for this problem. We will show with some examples that slight modifications of a minimizing k-partition in a proper way (preserving the value of the maximum relative diameter) will produce minimizing k-partitions as well, and so we do not have uniqueness in this setting (as in most of the problems involving the diameter functional). This subsection concludes by posing some necessary conditions for a given k-partition to be minimizing, derived from Lemma 3.2 and Theorem 4.5. In Section 5 we study the optimal bodies for this problem. More precisely, for each k∈N,k⩾3, we characterize the k-rotationally symmetric planar convex body for fixed area attaining the lowest possible value for the maximum relative diameter when considering k-partitions. Notice that, in view of Theorem 4.5, we have to focus on the standard k-partitions, and as a consequence of Lemma 3.2 we are able to prove our Theorem 5.1: MINIMIZING SUBDIVISIONS FOR THE MAXIMUM RELATIVE DIAMETER 5 For any k∈N,k⩾3, the minimum value for the maximum relative diameter for k-partitions in the class of k-rotationally symmetric planar convex bodies is uniquely attained by the standard k-partition associated to the intersection of the unit circle and the regular k-gon with inradius equal to 1/(2 sin(π/k)), up to dilations. The previous Theorem 5.1 characterizes the optimal body, for each k∈N, k⩾3, as the intersection of the unit circle with a particular regular k-gon. In Subsection 5.1 we analyze these intersections in each case, obtaining the precise optimal sets: for k= 3 and k= 5, the optimal bodies are respectively an equilateral triangle and a regular pentagon with rounded vertices, for k= 4 we have a square, and for k⩾6 the best set in this setting is the unit circle. We remark that the optimal body for k= 3 was already characterized in [3, Th. 4.7] by means of a constructive procedure. Figure 2. Optimal sets for k= 3, k= 4, k= 5, and k⩾6 Finally, in Section 6 we complete our study by considering large values of k∈N, discussing our problem for k-subdivisions (focusing on planar convex bodies). Recall that Theorem 4.5 establishes that the standard k-partitions are minimizing k-partitions for any k∈N,k⩾3, and minimizing k-subdivisions only when k⩽6, for any k-rotationally symmetric planar convex body. In Corollary 6.4 we obtain a lower bound for the maximum relative diameter functional when considering k-subdivisions of a planar convex body, involving a negligible addend when kis large enough. However, we think that such lower bound can be improved, and some reasonings in this section lead us to Conjecture 6.8: for large values of k∈N, a minimizing k-subdivision will be given by a decomposition induced by a planar tiling of regular hexagons. 2. Preliminaries In this paper we will consider, for any given k∈N,k⩾3, the class Ckof k-rotationally symmetric planar convex bodies (assuming therefore the compactness of the sets). We recall that, in this setting, each set Cin the class Ckhas a center of symmetry p, and then, the k-rotationally symmetry means that Cis invariant under the rotation centered at pwith angle ϕk:= 2π/k. Some examples of sets belonging to Ckare the regular k-gon and the Reuleaux k-gon. Given a planar compact set C, we shall denote by r(C) the inradius of C(that is, the radius of the largest ball contained in C), and by R(C) the 6 A. CA ˜ NETE, U. SCHNELL, AND S. SEGURA GOMIS Figure 3. Some examples of k-rotationally symmetric planar convex bodies: the equilateral and Reuleaux triangles are 3- rotationally symmetric, the regular octogon is 8-rotationally symmetric (and also 4-rotationally symmetric), and the circle is k-rotationally symmetric for any k∈N. circumradius of C(radius of the smallest ball containing C). If no confusion may arise, we shall simply denote them by rand R, respectively. For krotationally symmetric planar convex bodies, we have the following Lemma, which provides a metric characterization for the inradius and the circumradius. Lemma 2.1. Let C∈ Ck, and let pbe its center of symmetry. Then, the inradius rcoincides with the minimal Euclidean distance between pand a point in ∂C, and the circumradius Rcoincides with the maximal Euclidean distance between pand a point in ∂C. Proof. It is known that the set Iof centers of all largest balls contained in Cis a convex set with empty interior [13]. As Cis k-rotationally symmetric, this implies that I={p}(note that for any c∈I,c6=p, the images of cby the existing rotations would belong to I, and so pwould be an interior point of I). Then there is a unique largest ball contained in C, which is centered at p. This immediately yields the statement for the inradius. On the other hand, as the smallest ball containing Cis unique, its center has to coincide with p(otherwise, an analogous rotation argument will contradict that uniqueness). This implies trivially the statement for the circumradius.  R r Figure 4. The inradius rand the circumradius Rof an equilateral triangle. We now define the notions of k-subdivision and k-partition of a planar compact set, which will be considered along this work. Notice that these definitions can be generally done without assuming additional hypotheses regarding convexity or symmetry. MINIMIZING SUBDIVISIONS FOR THE MAXIMUM RELATIVE DIAMETER 7 Definition 2.2. Let Cbe a planar compact set. A k-subdivision Sof Cis a decomposition of Cinto kconnected subsets {C1,...,Ck}, not necessarily of equal areas, satisfying: i) C= k [ i=1 Ci, ii) int(Ci)∩int(Cj) = ∅, for i, j ∈ {1,...,k},i6=j. In addition, if a k-subdivision Sis given by kcurves meeting at a point c∈int(C), and the other endpoints of the curves are different points in ∂C, then Swill be called a k-partition of C. In this case, cis called the common point of the k-partition and the endpoints of the curves meeting ∂C will be called the endpoints of the k-partition. We stress that k-subdivisions are general decompositions of the original set Cinto kconnected subsets, and that k-partitions are k-subdivisions with a particular topological structure: they consist of kcurves converging to an interior point, each of them meeting the boundary of Cat a different point. Figure 5. Three different 5-subdivisions for the regular pentagon. Figure 6. Three different 5-partitions for the regular pentagon. Remark 2.3. In this work, we are focusing on the class Ckof k-rotationally symmetric planar convex bodies since the lack of the rotational symmetry prevents a general study of this question. Moreover, for any set C∈ Ck, its center of symmetry is a remarkable point in the interior of C, and so it seems natural to consider k-partitions, which are precisely the k-subdivisions determined by interior points of C. We now recall the definition of the maximum relative diameter for a ksubdivision of a planar compact set. Definition 2.4. Let Cbe a planar compact set, and let Sbe a k-subdivision of Cinto subsets {C1,...,Ck}. We define the maximum relative diameter associated to Sas dM(S, C) = max{D(Ci) : i= 1,...,k}, 8 A. CA ˜ NETE, U. SCHNELL, AND S. SEGURA GOMIS where D(Ci) = max{d(x, y) : x, y ∈Ci}denotes the Euclidean diameter of Ci. Remark 2.5. Along this work, we shall usually write dM(S) instead of dM(S, C), if no confusion may arise. Remark 2.6. For a given planar compact set Cand a k-subdivision Sof C, the existence of dM(S, C) is assured due to the continuity of the Euclidean distance d:R2→R, and the compactness of C. Remark 2.7. By a standard argument it is clear that dM(S, C) is attained by points lying in the boundary of one of the subsets Cidetermined by the k-subdivision S. Moreover, it is well-known that the diameter of a convex polygon is given by the distance between two of its vertices. In the forthcoming sections, we shall investigate the k-partitions and the k-subdivisions of a given k-rotationally symmetric planar convex body which minimize the maximum relative diameter functional. In other words, for a given set C∈ C, we shall look for min{dM(P, C) : Pis a k-partition of C}, and min{dM(S, C) : Sis a k-subdivision of C}. The k-partitions and k-subdivisions attaining those minimum values will be called minimizing. As usual in this kind of optimization problems involving the diameter functional, we point out that the uniqueness of the minimizing k-subdivision is not expected, as we shall see in Subsection 4.1. 3. Standard k-partition In this section we shall describe a particular k-partition for the sets of our class Ck, called the standard k-partition, by means of a simple geometrical intrinsic construction. This k-partitions will play an important role along this work. For C∈ Ck, let x1∈∂C be a point in ∂C with minimal distance to the center of symmetry p. In view of Lemma 2.1, this implies that d(p, x1) is the inradius of C, and so the segment p x1will be referred to as an inradius segment of C(notice that for regular polygons, such a segment is called apothem). By applying k−1 times the rotation centered at pwith angle ϕk= 2π/k to the point x1, we shall finally obtain a set of points {x1,...,xk} ⊂ ∂C, all of them minimizing the distance to p. If we join these points to p, we shall get kinradius segments dividing Cinto kconnected congruent subsets. This k-partition will be called standard k-partition of C, and will be denoted by Pk(C), or simply Pk. Remark 3.1. In general, the previous construction of the standard k-partition associated to a k-rotationally symmetric planar convex body Cis not uniquely determined. In fact, if there are more than kpoints in ∂C attaining the minimal distance to the center of symmetry of C, then the associated standard kpartition is not unique (for instance, this happens for the standard 3-partition MINIMIZING SUBDIVISIONS FOR THE MAXIMUM RELATIVE DIAMETER 9 Figure 7. Standard k-partitions for different k-rotationally symmetric planar bodies of the regular hexagon, or for the standard k-partition of the circle for any k⩾2). In any case, all the standard k-partitions associated to Cwill be k-rotationally congruent by construction. For the standard k-partition of a given set Cin Ck, the associated maximum relative diameter can be easily computed in terms of the inradius and the circumradius of C, as proved in the following Lemma 3.2. Lemma 3.2. Let Cbe a k-rotationally symmetric planar convex body, with inradius rand circumradius R. Let Pkbe the standard k-partition associated to C. Then, dM(Pk, C) = max{R, 2rsin(ϕk/2)}. Remark 3.3. Recall that ϕk/2 = π/k. Proof. We can assume that dM(Pk, C) = D(C1), where C1is one of the congruent subsets determined by Pk. Call x1, x2∈∂C1the endpoints of the two inradius segments bounding C1. Any pair of points realizing D(C1) cannot be contained in the relative interior of an inradius segment (in that case, the diameter could be clearly increased by considering some proper point along the segment). As the maximal distance between two points on ∂C ∩∂C1is d(x1, x2) = 2 rsin(π/k), and the maximal distance between pand ∂C equals R, the statement follows.  The following examples show that both possibilities from Lemma 3.2 may occur. Example 3.4. (Circle) Any circle is a k-rotationally symmetric planar convex body for any k∈N,k⩾2, whose inradius rcoincides with its circumradius R. Consequently, in view of Lemma 3.2, it is immediate checking that dM(Pk) = 2rsin(ϕk/2) for k∈ {3,4,5,6}, and dM(Pk) = Rfor k⩾6. Example 3.5. (Regular polygons) For a regular k-gon Ek, Lemma 3.2 yields that dM(Pk, Ek) = R(Ek), for k∈N,k⩾3, due to the well-known relation r(Ek) = R(Ek) cos(ϕk/2). Remark 3.6. Lemma 3.2 does not hold for k= 2 (it can be easily checked by considering the square or the circle). The reason of this fact is that the standard 2-partition P2will divide any centrally symmetric planar convex body C into two subsets whose boundaries do not contain the center of symmetry p as a vertex. Notice that pwill belong to the line segment x1x2(being x1, x2 16 A. CA ˜ NETE, U. SCHNELL, AND S. SEGURA GOMIS Figure 12. Optimal bodies for k= 3, k= 4, k= 5, and k⩾6. Remark 5.3. The optimal body for k= 3 was already obtained in [3, Th. 4.7] by means of a constructive procedure, and is a solution for some others geometric optimization problems, see [9, §. 4]. 6. k-subdivisions for large values of k In Section 4 we have proved that, for each k-rotationally symmetric planar convex body, the associated standard k-partition is a minimizing k-partition for the maximum relative diameter, for any k∈N,k⩾3 (Theorem 4.5), and also a minimizing k-subdivision for k∈ {3,4,5,6}(Theorem 4.6). In this section, we will go further with the study of the k-subdivisions problem, by considering the asymptotic situation for large values of k∈N, and searching for a minimizing k-subdivision for the maximum relative diameter. We have already seen that, in general, the standard k-partition is not minimizing when k⩾7 (for instance, see Examples 4.7 and 4.8). The reasonings in this section will lead us to conjecture that an optimal k-subdivision, when the number of regions kis large enough, is provided by a hexagonal tiling (up to some subsets touching the boundary of the original set). This kind of tilings have already appeared in the literature during the last years, specially from the isoperimetric point of view (see [10] for the least-perimeter subdivision of the plane, or [6] for the structure of the solutions of the planar soap bubble problem for a large number of regions). We point out that the assumption of rotational symmetry can be removed, and so we shall focus on general planar convex bodies. The following lemma, whose proof relies on the classical isodiametric inequality [2], provides a first lower bound for the maximum relative diameter, when considering k-subdivisions of an arbitrary planar convex body. Lemma 6.1. Let Cbe a planar convex body, and let Sbe a k-subdivision of C, for k∈N. Then, (6.1) dM(S, C)⩾rA(C) kr4 π. Proof. Let C1,...,Ckbe the subsets of Cgiven by S, and call d=dM(S, C). By using the isodiametric inequality [2], since D(Ci)⩽d, we have that MINIMIZING SUBDIVISIONS FOR THE MAXIMUM RELATIVE DIAMETER 17 A(Si)⩽π 4d2, for i= 1,...,k. Then, A(C) = k X i=1 A(Ci)⩽kπ 4d2, which yields the statement.  Remark 6.2. We recall that the equality in the classical isodiametric inequality holds for circles, and so, the equality in (6.1) cannot be attained, since it is not possible to consider a subdivision given by circles due to the overlappings of the corresponding subsets. Therefore, (6.1) actually provides a strict lower bound for the maximum relative diameter functional. The following Lemma 6.3 will allow to improve the lower bound obtained in the previous Lemma 6.1, for large values of k. Lemma 6.3. Let Cbe a planar convex body, and let Sbe a k-subdivision of Cwith dM(S, C) = d. Then, (6.2) A(C)⩽0.688452 k d2+d P(C), where P(C)denotes the perimeter of C. Proof. Call C1,...,Ckthe subsets of Cgiven by S. Let f Cibe the convex hull of Ci. Then D(f Ci) = D(Ci) for i= 1,...,k, and Cis covered by the union of these convex hulls. Since the possible overlappings can be eliminated by dividing the intersection regions by proper line segments, we can then assume that C1,...,Ckare convex polygons up to some arcs contained in ∂C. By replacing each of these arcs by the line segment determined by its endpoints, we shall obtain convex polygons C′ i⊂Ci,i= 1,...,k, whose union is a convex polygon C′contained in C. It is clear that C\C′is contained in the union of stripes parallel to ∂C′and breadth d. This implies (6.3) A(C)⩽A(C′) + d P(C′)⩽A(C′) + d P (C) = k X i=1 A(C′ i) + d P(C). The convex polygons C′ 1,...,C′ kdetermine a planar graph with k+ 1 faces (together with the unbounded component C′ 0). If mand ndenote the number of edges and vertices, by Euler’s theorem we have (k+ 1) −m+n= 2. Since each edge is incident to two vertices and the degree of each vertex is at least 3 we obtain 2m⩾3n. It follows 1 = k−m+n⩽k−m+2/3m=k−m/3 and so m⩽3k−3. Let fjbe the number of polygons with jedges in {C′ 1,...,C′ k}, and sthe number of edges of C0. Then 2m=s+X j fjj 18 A. CA ˜ NETE, U. SCHNELL, AND S. SEGURA GOMIS and so X j fjj < s +X j fjj= 2 m⩽6k−6⩽6k. On the other hand, let mjbe the maximal area of a polygon of jedges with diameter one. It is known (see [12]) that mjis attained by regular polygons for odd j, with mj=j/2 cos(π/j) tan(π/(2j)). Furthermore, the values for m4,m6and m8are also numerically known (see [12]), so that m3= 0.433012, m4= 0.5, m5= 0.657163, m6= 0.6749814429, m7= 0.719740, m8= 0.7268684828, m9= 0.745619. Moreover, notice that mj⩽π/4, for any j⩾3, as the corresponding polygon will be contained in the circle of diameter one. Then, if C′ iis a polygon with jedges, A(C′ i)⩽d2mj, and so (6.4) k X i=1 A(C′ i)⩽X j fjmjd2=d2X j fjmj⩽d29 X j=3 fjmj+π/4X j⩾10 fj. We are interested in finding an upper bound for (6.4). By replacing Pj⩾10 fj by e fand calling em=π/4, and using linear programming, it is straightforward checking that the maximal value for f3m3+... +f9m9+e femunder the restrictions f3+...+f9+e f=k, 3f3+...+9f9+10 e f⩽6k, and f3,...,f9,e f⩾0 is given by k(m5+m7)/2 = 0.688452 k, being attained by f5=f7=k/2 and f3=...=f9=e f= 0. Using that bound in (6.4), we conclude that Pk i=1 A(C′ i)⩽0.6884 k d2, which yields the statement taking into account (6.3).  Corollary 6.4. Let Cbe a planar convex body, and let Sbe a k-subdivision of C. Then (6.5) dM(S, C)⩾rA(C) kr1 0.688452 −O(1/k). Proof. The statement is a direct consequence of (6.2).  Remark 6.5. For large values of k∈N, (6.5) gives an improvement of the lower bound in (6.1). However, such a bound is strict, since a planar tiling by regular pentagons and regular heptagons cannot be realized. In the following Lemma 6.6 we shall consider a particular k-subdivision for any planar convex body and any k∈N, induced by a proper tiling of the plane by regular hexagons, see Figure 13. We shall estimate the maximum relative diameter dMfor these k-subdivisions, obtaining an upper bound for the minimal value of dM, which we conjecture it is the optimal one for large values of k∈N. Lemma 6.6. Let Cbe a planar convex body C. For each k∈N, there exists ak-subdivision Sksuch that (6.6) dM(Sk, C)⩽rA(C) ks8 3√3+O1 k. MINIMIZING SUBDIVISIONS FOR THE MAXIMUM RELATIVE DIAMETER 19 Proof. Call Hthe regular hexagon with unit diameter, and so A(H) = 3√3/8. For any d > 0, let Ldbe the planar lattice with fundamental cell d H. For any k∈N, let dkbe the unique positive solution of (6.7) k A(d H) = A(C) + d P(C) + π d2, where P(C) denotes the perimeter of C. It is clear that Cis covered by the union of the fundamental cells of Ldkwith nonempty intersection with C. Call nthe number of such fundamental cells, which will be contained in the outer parallel body C+dkB, where Bis the planar unit ball. Then, Steiner’s formula and (6.7) give n A(dkH)⩽A(C+dkB) = A(C) + dkP(C) + π d2 k=k A(d H), and so n⩽k. This means that the lattice Ldkinduces a k-subdivision Skof C(if n < k, we can divide a fundamental cell until having a decomposition of Cinto kconnected subsets) with dM(Sk, C)⩽dk. This fact, together with a straightforward estimate of dkby using (6.7), proves the statement.  Remark 6.7. We stress that the k-subdivision Skfrom Lemma 6.6 consists of regular hexagons of diameter dkin the interior of the set, and portions of such hexagons when meeting the boundary of the set. Figure 13. A subdivision of the circle given by regular hexagons The previous Lemma 6.6 gives an upper bound for the minimum possible value of the maximum relative diameter for large enough values of k∈N. However, we think that such a bound is the optimal one due to the following fact: taking into account Lemma 6.1 and using the covering density [8] (recall that the subdivision by circles would produce overlappings of the subsets), it is easy to check that the bound from (6.6) can be obtained from the bound given in (6.1), when kis large enough. This means, in some sense, that the hexagonal tiling corrects the defects appearing in the decomposition by circles. In view of this, we finish this section with the following Conjecture 6.8, regarding the minimizing k-subdivisions for a planar convex set and large values of k∈N. Conjecture 6.8. Let Cbe a planar convex set, and let Sbe any k-subdivision of C. For large enough values of k∈N, the minimal value for dM(S, C) is provided by the k-subdivision described in Lemma 6.6, given by regular hexagons of fixed diameter in the interior of Cand portions of such hexagons when meeting the boundary of C. 20 A. CA ˜ NETE, U. SCHNELL, AND S. 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MINIMIZING SUBDIVISIONS FOR THE MAXIMUM RELATIVE DIAMETER 21 Departamento de Matem´ atica Aplicada I, Universidad de Sevilla E-mail address:[email protected] Faculty of Mathemics and Sciences, University of Applied Sciences Zittau/G¨ orlitz E-mail address:[email protected] Departamento de An´ alisis Matem´ atico, Universidad de Alicante E-mail address:salvador.seg[email protected]