Bisections of centrally symmetric planar convex bodies minimizing the maximum relative diameter
aa r X i v : . [ m a t h . M G ] M a r BISECTIONS OF CENTRALLY SYMMETRIC PLANARCONVEX BODIES MINIMIZING THE MAXIMUMRELATIVE DIAMETER
ANTONIO CA ˜NETE AND SALVADOR SEGURA GOMIS
Abstract.
In this paper we study the bisections of a centrally symmetricplanar convex body which minimize the maximum relative diameter func-tional. We give necessary and sufficient conditions for being a minimizingbisection, as well as analyzing the behavior of the so-called standard bi-section. Introduction
Historically, the classical geometric functionals (perimeter, area, volume,inradius, circumradius, diameter and width), and the relations between them,have been intensely studied, yielding a great variety of optimization prob-lems [14, 8]. Possibly, the most relevant example is the isoperimetric problem ,examining the relation between the area and the volume of sets in R n [13, 12].Moreover, in the setting of Convex Geometry, these functionals play an im-portant role and can be considered as the origin of this theory.In this context, we shall focus on a particular relative geometric problemconcerning the diameter functional. This is one of the most natural magni-tudes for measuring the size of a set, and has been deeply considered in theliterature. Some well-known important results in R n regarding this functionalare, for instance, Jung’s theorem [9], establishing the inequality between thediameter and the circumradius of a compact set, the isodiametric inequality [1], which asserts that the ball is the compact convex set of fixed volume withthe minimum possible diameter, or
Borsuk’s conjecture [2], asking whetherany compact set K can be divided into n + 1 subsets whose diameters arestriclty less than the diameter of K . Additionally, more inequalities involvingthe diameter and other classical functionals for planar compact convex setscan be found in [14].In this work we shall consider the maximum relative diameter functional in R , which is defined in the following way: for a fixed planar compact convexset C , a division of C into two connected subsets determined by a simple curvewith endpoints in the boundary of C will be called a bisection of C . Then,given a bisection P of C with subsets C , C (not necessarily enclosing equal Mathematics Subject Classification.
Key words and phrases.
Centrally symmetric planar convex bodies, maximum relativediameter. areas), the maximum relative diameter associated to P is d M ( P ) = max { D ( C ) , D ( C ) } , where D ( S ) denotes the Euclidean diameter of a planar set S . In view ofthis definition, the maximum relative diameter clearly represents the largestdistance in the subsets generated by the bisection. In this setting, we areinterested in finding the optimal bisection for the maximum relative diameterfunctional. That is, among all the bisections of C , we look for the one attainingthe minimum possible value for this functional, which can be considered as thedivision of C with both subsets as small as possible in terms of the diameter.Throughout this paper, we shall assume that our sets are centrally sym-metric. This hypothesis provides enough geometric structure to deal with thisproblem, allowing to obtain descriptive results for the minimizing bisections(in the non-symmetric case, it seems not possible to find similar properties forthe optimal bisections, due to the lack of symmetry). Moreover, consideringdivisions of the sets into two subsets is something naturally inherent to centralsymmetry. On the other hand, we point out that if we focus on bisections bystraight lines in the class of compact convex sets, the minimum value for themaximum relative diameter is precisely attained by a centrally symmetric set[11, Th. 7] so, in some sense, this kind of sets is certainly suitable for thisfunctional.Our main results refer to the minimizing bisections for the maximum rel-ative diameter of centrally symmetric planar compact convex sets. We shallsee that we do not have uniqueness of solution for this problem, since properslight modifications of a minimizing bisection will be also minimizing (this isa common feature when working with the diameter functional). Moreover,in order to find a minimizing bisection, our Proposition 3.1 assures that it isenough to focus on the bisections given by a straight line passing through thecenter of symmetry of the set (which will always generate symmetric subsetsenclosing equal areas). This property agrees with the intuitive idea that thecorresponding subsets of an optimal bisection must be as balanced as possible.Proposition 3.4 shows a necessary condition for a bisection of this type tobe minimizing (expressed in terms of the farthest distances from its two end-points), which is complemented in our Theorem 3.6, establishing a criterionfor asserting that a bisection (by a straight line passing through the center ofsymmetry) is minimizing.This partitioning optimization problem has been already considered in [11],but with the additional restriction of bisections with equal-area subsets . In thissetting, among other results, it is proved that a minimizing bisection is alwaysgiven by a straight line passing through the center of symmetry of the set [11,Prop. 4], with no further description of the properties of the solutions. Ourwork is inspired in this paper, with the aim of extending the results thereinto a more general situation (arbitrary bisections with non-equal area subsets),and describing the minimizing bisections in a more precise way.It is worth mentioning that the analogous question for divisions into a largernumber of subsets has been also studied: for a given a k -rotationally symmetric INIMIZING BISECTIONS FOR THE MAXIMUM RELATIVE DIAMETER 3 planar compact convex set C , with k ∈ N , k >
3, a k -partition of C is adecomposition of C into k connected subsets C , . . . , C k , determined by k simple curves starting in ∂C , all of them meeting in an interior point of C .And similarly, given a k -partition P of C , we can define the maximum relativediameter of P as d M ( P ) = max { D ( C i ) : i = 1 , . . . , k } . In this context, we can investigate the k -partitions of C attaining the minimumvalue for d M . This was treated in [5] (see also [4]), where it is proved thatthe so-called standard k -partition (constructed by using k inradius segmentssymmetrically placed, see Figure 1) is a solution for this problem, for any k > Figure 1.
Some standard k -partitions for k > k = 2): a standard bisection (consisting of twosymmetric inradius segments) is not minimizing in general (for instance, seeExample 4.2). In fact, the standard bisection of a given centrally symmetricplanar compact convex set could not even be uniquely defined, as shown inExample 4.5. These are two remarkable differences with respect to the case of k -rotationally symmetric planar compact convex sets, with k > A. CA ˜NETE AND S. SEGURA GOMIS Preliminaries
Let us denote by C the class of centrally symmetric planar convex bodies(recall that a body is, as usual, a compact set). The central symmetry of a set C ∈ C means that there exists a point p ∈ C (called the center of symmetry of C ) such that C is invariant under the action of the rotation of angle π centered at p . Some examples of sets of this class are depicted in Figure 2. Figure 2.
Some centrally symmetric planar convex bodiesThroughout this paper, we shall focus on some particular divisions of oursets, called bisections . In Remark 2.5 we shall justify that these are the mostconvenient divisions for our problem. Note that the following definition canbe done in a more general setting.
Definition 2.1.
Let C ∈ C . A bisection of C is a decomposition of C intotwo connected subsets, given by a simple curve with endpoints in the boundary ∂C of C . Remark 2.2.
We point out that the curve determining a given bisection doesnot contain, in general, the center of symmetry of the set, and moreover,the corresponding subsets do not enclose necessarily equal areas, as shown inFigure 3.
Figure 3.
Three different bisections for an ellipseWe now proceed to define the geometric functional considered in this work,previously introduced in [11].
Definition 2.3.
Let C ∈ C , and let P be a bisection of C , with associatedsubsets C , C . The maximum relative diameter of P is defined as d M ( P ) = max { D ( C ) , D ( C ) } , where D ( S ) denotes the Euclidean diameter of S . Remark 2.4.
Recall that the diameter of a planar compact set is alwaysattained by a pair of points lying in the boundary of the set, and in the caseof a polygon, by two of its vertices.
INIMIZING BISECTIONS FOR THE MAXIMUM RELATIVE DIAMETER 5
For a fixed centrally symmetric planar convex body C , the purpose of thesenotes is investigating the bisections of C that minimize the maximum relativediameter functional, in the same spirit as in [4, 5], see also [11]: determin-ing these bisections precisely or, at least, describing some of their geometricalproperties. These bisections will be called minimizing along this paper. Inthis direction, some partial results have been obtained in the case of bisec-tions providing equal-area subsets : in this more restrictive setting, it has beenproved that, for any set C ∈ C , there always exists a minimizing bisectiongiven by a straight line passing through the center of symmetry of the set [11,Prop. 4]. However, no additional details have been outlined for the solutions,and nothing else is known. We shall consider this problem in the most gen-eral setting (that is, for bisections generating subsets which do not enclosenecessarily the same quantity of area), progressing in the description of theseoptimal bisections. Remark 2.5.
We point out that a given set C ∈ C can be decomposed intotwo connected subsets by means of divisions which are not bisections. Thiscan be done by using a simple closed curve entirely contained in the interior of C . In general, these decompositions are not good candidates for our problemsince, in view of Remark 2.4, all of them have maximum relative diameter equalto D ( C ), which is an immediate upper bound for our functional. Therefore,they will not be taken into account in these notes, and we shall focus on thenotion of bisection from Definition 2.1. Remark 2.6.
The uniqueness of solution is not expected for this optimizationproblem, as it usually occurs for questions involving the diameter functional.In fact, if we have a minimizing bisection P of a centrally symmetric planarconvex body C , slight modifications of P can be done preserving the valueof the maximum relative diameter, being minimizing as well. This propertysuggests that a complete description of all the minimizing bisections of C isnot a feasible task.2.1. Bisections by a straight line passing through the center of sym-metry.
Let C ∈ C , and let p be the center of symmetry of C . The bisectionsof C given by a straight line passing through p possess some special propertiesand will play an important role for our problem. Notice that for a bisection P of this type, the corresponding subsets C , C will be congruent due to the ex-isting symmetry (they will coincide up to the rotation of angle π about p ), andso both of them will enclose the same quantity of area, and D ( C ) = D ( C ).Denoting by v , v ∈ ∂C the endpoints of the line segment determining P ,[11, Prop. 3] leads to(2.1) d M ( P ) = max { d ( v , x ) : x ∈ ∂C } , where d stands for the Euclidean distance in the plane. Equality (2.1) impliesthat the maximum relative diameter of P will be given by the distance betweenan endpoint of P and any of its corresponding farthest points in ∂C .Apart from this, for a bisection P determined by a straight line passingthrough p , there is another equivalent expression for computing d M ( P ), which A. CA ˜NETE AND S. SEGURA GOMIS will be useful along this work. For any x ∈ ∂C , we shall denote by F C ( x )the set of farthest points from x in ∂C (notice that F C ( x ) is non-empty dueto compactness, and it may reduce to a single point). Since D ( C ) = D ( C ),we can just focus on one of the subsets provided by P , and using again [11,Prop. 3], we will have that(2.2) d M ( P ) = D ( C ) = max { d ( v , φ C ( v )) , d ( v , φ C ( v )) } , where v , v are the endpoints of P , and φ C ( v i ) ∈ F C ( v i ), i = 1 , Remark 2.7.
We note that it is easy to check that equalities (2.1) and (2.2)are not true neither for bisections given by a general planar curve, nor by astraight line which does not pass through the center of symmetry, see Figure 4.
Figure 4.
Equalities (2.1) and (2.2) do not hold for these twobisections of the ellipse
Remark 2.8.
For a given C ∈ C , and an arbitrary bisection P of C (notnecessarily determined by a straight line), with endpoints v , v ∈ ∂C , itmay happen that v ∈ F C ( v ) and v ∈ F C ( v ). In that case, it turns that d M ( P ) = d ( v , v ), in view of (2.2), and moreover, C will be contained inthe symmetric lens B ( v , d M ( P )) ∩ B ( v , d M ( P )), where B ( x, r ) denotes theEuclidean ball with center x and radius r .3. Main results
In this section we obtain the main results of this paper. Proposition 3.1,which is an extension of [11, Prop. 4], shows that there is always a minimizingbisection given by a straight line passing through the center of symmetryof the set. Proposition 3.4 states a necessary condition for a bisection to beminimizing, and Theorem 3.6 establishes some conditions which allow to assertthat a given bisection is minimizing. These last two results (which are provedfor bisections given by a straight line passing through the center of symmetry)reveal some of the geometric restrictions for being optimal.
Proposition 3.1.
Let C ∈ C , and let p be the center of symmetry of C . Let P be a minimizing bisection for d M (whose subsets do not enclose necessarilyequal areas). Then there exists a bisection P ′ given by a straight line passingthrough p such that d M ( P ) = d M ( P ′ ) .Proof. Let C , C be the subsets determined by P , and let v , v be theendpoints of P (notice that v ∈ C ∩ C ). We can assume that d M ( P ) = D ( C ) > D ( C ). Let v ′ ∈ ∂C be the symmetric point of v with respect INIMIZING BISECTIONS FOR THE MAXIMUM RELATIVE DIAMETER 7 to p , and consider the bisection P ′ given by the segment v v ′ (which passesthrough p ).Taking into account (2.1), we have that d M ( P ′ ) = d ( v , z ), for certain z ∈ ∂C . If z ∈ ∂C , then d ( v , z ) D ( C ) = d M ( P ). And if z ∈ ∂C , then d ( v , z ) D ( C ) D ( C ) = d M ( P ). Thus d M ( P ′ ) = d ( v , z ) d M ( P ),which implies that d M ( P ′ ) = d M ( P ) since P is minimizing. (cid:3) Remark 3.2.
A consequence of Proposition 3.1 is that, in order to find aminimizing bisection for a centrally symmetric planar convex body, we canfocus on bisections given by a straight line passing through the correspondingcenter of symmetry. Note that for these bisections, the endpoints are always symmetric with respect to the center of symmetry of the set.
Remark 3.3.
In fact, Proposition 3.1 shows that, if C ∈ C and p denotesits center of symmetry, for any bisection P of C we can find another bisection P ′ , given by a straight line passing through p , with d M ( P ′ ) d M ( P ).Next Proposition 3.4 states a necessary condition for a bisection (given bya straight line passing through the center of symmetry, in view of Remark 3.2)to be minimizing, by means of the farthest points of the endpoints of thebisection. In some sense, this result suggests a certain balance for the optimaldivisions: the distances between each endpoint and its corresponding farthestpoint must coincide (being also equal to the value of the maximum relativediameter, due to equality (2.2)). Proposition 3.4. (Necessary condition) Let C ∈ C , and let p be the centerof symmetry of C . Let P be a bisection of C given by a straight line pass-ing through p , with endpoints v , v ∈ ∂C , and subsets C , C . If P is aminimizing bisection for d M , then (3.1) d M ( P ) = d ( v , φ C ( v )) = d ( v , φ C ( v )) , where φ C ( v i ) ∈ F C ( v i ) .Proof. Assume that d M ( P ) = D ( C ) = d ( v , φ C ( v )) > d ( v , φ C ( v )). Byapplying a slight rotation centered at p to the straight line containing thesegment v v , it is clear that we can consider a new bisection e P , with newendpoints e v , e v ∈ ∂C and subsets f C , f C , satisfying that e v ∈ ∂C . Byconstruction, we shall have that d ( v , φ C ( v )) > d ( e v , φ C ( v )). Due to thecontinuity of the Euclidean distance, as e v i is close to v i , i = 1 ,
2, it followsthat the inequality d ( e v , φ f C ( e v )) > d ( e v , φ f C ( e v )) will be preserved, at least,infinitesimally. This implies that d M ( e P ) = d ( e v , φ f C ( e v )), by using (2.2).Moreover, d ( e v , φ f C ( e v )) will be close to d ( e v , φ C ( v )). Thus, d M ( e P ) = d ( e v , φ f C ( e v )) < d ( v , φ C ( v )) = d M ( P ) , which contradicts the minimizing character of P . (cid:3) Remarks 3.5.
We shall mention some brief comments concerning Proposi-tion 3.4.
A. CA ˜NETE AND S. SEGURA GOMIS i) The reverse of Proposition 3.4 does not hold in general: this can beseen by considering, for instance, a rectangle and the bisection given bythe orthogonal line to the shortest edges passing through the center ofsymmetry, which satisfies (3.1) but it is clearly not minimizing. There-fore, some additional hypotheses are needed for an eventual sufficientcondition.ii) A geometric interpretation of this result is that the maximum relativediameter of a minimizing bisection (given by a straight line passingthrough the center of symmetry) is necessarily provided by at least two different segments in each congruent subset, unless it is uniquelyachieved by the distance between the endpoints of the bisection.iii) The reader may compare Proposition 3.4 with [11, Prop. 3]: in the caseof a minimizing bisection given by a straight line passing through thecenter of symmetry, the farthest distances from both endpoints mustcoincide, providing the value of the maximum relative diameter.We shall now prove our main Theorem 3.6, which establishes some condi-tions to assert that a given bisection is minimizing.
Theorem 3.6.
Let C ∈ C , and let p be the center of symmetry of C . Let P be a bisection of C given by a straight line passing through p , with endpoints v , v , and subsets C , C . If there exist φ C ( v ) ∈ F C ( v ) , φ C ( v ) ∈ F C ( v ) such that i) d ( v , φ C ( v )) = d ( v , φ C ( v )) , and ii) ∂C ⊂ A ∪ A , where A i is the complement in the plane of the Eu-clidean ball B i = B ( φ C ( v i ) , d ( v i , φ C ( v i ))) , for i = 1 , ,then P is a minimizing bisection for d M .Proof. Notice that d M ( P ) = d ( v , φ C ( v )) = d ( v , φ C ( v )), in view of (2.2)and the assumed hypothesis. Consider now any bisection e P of C determinedby a straight line passing through p , with endpoints e v , e v . One of these end-points, say e v , will necessarily lie in ∂C , and so e v ∈ A ∪ A . Without loss ofgenerality, we can assume that e v ∈ A . Then, d ( e v , φ C ( v )) > d ( v , φ C ( v )),and so d M ( e P ) > d ( e v , φ C ( v )) > d ( v , φ C ( v )) = d M ( P ) , which yields the minimizing character of P , taking into account Proposi-tion 3.1. (cid:3) Remarks 3.7.
Regarding Theorem 3.6, we point out the following comments:i) The second hypothesis is equivalent to ∂C ∩ ( B ∩ B ) = ∅ , with thenotation therein.ii) It is not difficult to check that the second hypothesis implies that d M ( P ) = d ( v , v ).iii) It may happen that F C ( v ) ∩ F C ( v ) is a non-empty set. In that case,Theorem 3.6 can be applied trivially and the corresponding bisectionis minimizing. INIMIZING BISECTIONS FOR THE MAXIMUM RELATIVE DIAMETER 9
Remark 3.8.
We stress that, in order to apply Theorem 3.6, we need tofind appropriate farthest points from the endpoints of the bisection. That is,the hypotheses of Theorem 3.6 may not hold for all possible choices for thecorresponding farthest points, as shown in the following example. Considera rhombus C formed by joining two congruent equilateral triangles, and thebisection given by the common edges, with endpoints v , v ∈ ∂C , see Figure 5.It is clear that v ∈ F C ( v ) and v ∈ F C ( v ), but Theorem 3.6 cannot be usedwith those elections. However, the vertex q of C belongs to F C ( v ) ∩ F C ( v ),and so it is possible to apply the result for that farthest point, see Remarks 3.7,iii). q v v Figure 5.
A rhombus C formed by joining two congruent equi-lateral triangles, and a minimizing bisection of C Previous Theorem 3.6 is not sharp, in the sense that there exist minimizingbisections which cannot be identified by means of this result. This can beeasily seen for a circle: any bisection given by a diameter is optimal, but thesecond hypothesis is not verified (observe Remarks 3.7, ii)). We shall alsoillustrate this fact in the following Example 3.9.
Example 3.9.
Let C be the centrally symmetric hexagon depicted in Fig-ure 6, obtained by cutting symmetrically two opposite corners of a square (thelengths of the resulting edges are 3 .
31 and 5 .
66 units, with non-right anglesequal to 3 π/ P determined by the segment joiningthe midpoints of the shortest edges of C . It can be checked that d M ( P ) = 8 . v and φ C ( v ), see Figure 7.For any other bisection P ′ given by a straight line passing through the centerof symmetry, it follows that d M ( P ′ ) > d M ( P ), since one of the correspondingsubsets will contain the segment v φ C ( v ) or v φ C ( v ), both with lengthequal to d M ( P ), or the segment x φ C ( v ) or x φ C ( v ), both with length equalto 8 .
34 (where φ C ( v i ) is the farthest point from v i in ∂C , i = 1 , P is a minimizing bisection, but Theorem 3.6 cannot be applied becausethe second hypothesis does not hold, as shown in Figure 7.4. Standard bisection
In this section we shall introduce a particular bisection for a centrally sym-metric planar convex body, which is called standard bisection . Its construction
Figure 6.
Centrally symmetric hexagon obtained by cutting a square
Figure 7.
Theorem 3.6 cannot be applied since two pieces of ∂C are not contained in A ∪ A is analogous to the one described in [5, § . 3], which concerns the standard k -partitions of k -rotationally symmetric planar convex bodies (for k ∈ N , k > Definition 4.1.
Let C ∈ C . A standard bisection of C is a decomposition of C determined by two symmetric inradius segments of C . We shall denote itby P ( C ) , or simply P . Note that, for a given set in C , it is always possible to construct an associ-ated standard bisection (due to the existing symmetry), which will consist ofa line segment passing through the center of symmetry. In fact, it will be oneof the shortest chords of the set passing through that point, see Figure 8.As indicated in the Introduction, it is known [5, Th. 4.5] that, for any k -rotationally symmetric planar convex body, its corresponding standard k -partition (defined by means of k inradius segments symmetrically placed) isalways minimizing for the maximum relative diameter functional, when k > INIMIZING BISECTIONS FOR THE MAXIMUM RELATIVE DIAMETER 11 p p
Figure 8.
Standard bisections for an ellipse and a rectanglesee Figure 1. It is then natural to wonder whether the standard bisectionis minimizing in our centrally symmetric case (which corresponds to k = 2).This holds for a wide variety of sets of our class, but it is not true in general,as shown in the following Example 4.2. Example 4.2.
Let C be a rhombus, and consider an associated standardbisection P of C , depicted in the left-hand side of Figure 9. It is clear that P is not minimizing, since the bisection P determined by the vertical linesegment passing through the center of symmetry (right-hand side of Figure 9)has smaller value for d M . In fact, P does not satisfy the necessary conditionfrom Proposition 3.4, and Theorem 3.6 yields that P is a minimizing bisectionof C . Figure 9.
The standard bisection P of the rhombus is not minimizingOne might think that the standard bisection from Example 4.2 is not min-imizing essentially because the necessary condition from Proposition 3.4 doesnot hold. The following example shows that even when this necessary condi-tion is satisfied, we cannot assure the minimizing character of a given standardbisection. Example 4.3.
Let S be a square, and call v , v the midpoints of the up-per and lower edges, and w , w the midpoints of the other two edges, seeFigure 10. Consider C = S ∩ B ( v , d ( v , v )) ∩ B ( v , d ( v , v )), which is acentrally symmetric planar convex body. It is clear that the bisection P of C provided by the segment v v is standard, as well as the bisection P ′ givenby w w . Both of them satisfy the necessary condition from Proposition 3.4,but we have that d M ( P ′ ) = d ( w , x ) > d ( v , v ) = d M ( P ), and so P ′ is notminimizing. Moreover, Theorem 3.6 implies that P is a minimizing bisectionfor d M .The two previous Examples 4.2 and 4.3 reveal that a standard bisection isnot optimal in general. Although some partial results can be obtained in somerestrictive situations, we shall refer to Theorem 3.6 in order to determine if agiven one is minimizing. Figure 10. P and P ′ are two standard bisections of C . Wehave that P is minimizing, while P ′ is not Remark 4.4.
Consider C ∈ C and a standard bisection P of C , with end-points v , v ∈ ∂C . If d M ( P ) = d ( v , v ), then P is necessarily minimizing(recall that, in this case, we cannot apply Theorem 3.6, see Remarks 3.7, ii)).The reason is that for any other bisection P determined by a straight linepassing through the center of symmetry p ∈ C , with endpoints w , w ∈ ∂C ,it follows that d ( w , w ) > d ( v , v ), since p v i is an inradius segment of C , i = 1 ,
2, and so d M ( P ) > d ( w , w ) > d ( v , v ) = d M ( P ). This property doesnot hold for bisections which are not standard: if we consider an ellipse C ,and the bisection P determined by the segment v v , where D ( C ) = d ( v , v ),then we clearly have that P is not minimizing, although d M ( P ) = d ( v , v ).4.1. Uniqueness of the standard bisection.
In general, the standard bi-section of a centrally symmetric planar convex body is not uniquely defined:we clearly have two different ones for a given square (joining the midpoints ofeach pair of opposite edges), and an infinite amount of them for a circle (pro-vided by the diameter segments). In these two cases, the maximum relativediameter of the different standard bisections coincide, and so this fact is notrelevant for our optimization problem. However, the lack of uniqueness mayalso refer to the values of the maximum relative diameter, as shown in thefollowing Example 4.5.
Example 4.5.
Let C be a planar cap body , that is, the convex hull of acircle and two exterior symmetric points with respect to the center (whichwill be called the vertices of C ). This centrally symmetric planar convex bodypossesses an infinite quantity of associated standard bisections, determinedby each pair of symmetric points lying in the circular pieces of ∂C . In thissetting, if the vertices of C are far enough from the center of the circle, all thestandard bisections of C will have different values for the maximum relativediameter. For instance, for the two standard bisections from Figure 11, themaximum relative diameter equals the distance between an endpoint of thebisection and a vertex of the cap body, thus attaining distinct values. We INIMIZING BISECTIONS FOR THE MAXIMUM RELATIVE DIAMETER 13 point out that the same happens for the standard bisections of the set fromExample 3.9, as indicated in Section 5 below.
Figure 11.
Two standard bisections with different values for d M Remark 4.6.
The behavior described in Example 4.5 is another remarkablepeculiarity of our problem with respect to the analogous one for k -rotationallysymmetric planar convex bodies ( k > k -partitionsalways yield the same value for the maximum relative diameter, due to [5,Lemma 3.2]. Remark 4.7.
For a given set C in C , the standard bisection of C is uniquelydefined if and only if the associated inball touches ∂C only twice.5. Some Examples
In this section we collect several examples of centrally symmetric planarconvex bodies, indicating one of the minimizing bisections in each case.The corresponding standard bisections are minimizing for the square, therectangle or the ellipse, by direct application of Theorem 3.6.The case of the circle is special, since the maximum relative diameter func-tional is constant for any arbitrary bisection (such a constant is the diameterof the circle). Therefore, any bisection of the circle can be considered mini-mizing. pp p
Figure 12.
Minimizing bisections for the rectangle, the ellipseand the circleFor the hexagon treated in Example 3.9, which is depicted in Figure 6, wehave already described a minimizing bisection. We point out that such a bi-section is standard, and that there are two other standard bisections for thisset (joining each pair of larger opposite symmetric edges), which are not min-imizing (it can be checked that the necessary condition from Proposition 3.4does not hold). Recall that Theorem 3.6 cannot be applied for this optimalbisection.
We have studied the rhombus in Example 4.2. The standard bisection isnot minimizing, and Theorem 3.6 yields that the bisection given by a verti-cal straight line passing through the center of symmetry minimizes d M , seeFigure 9.For the cap body from Example 4.5, we have already indicated that thereare an infinite amount of associated standard bisections, each of them with adifferent value for the maximum relative diameter when the vertices are farenough from the center of symmetry, see Figure 11. In these situations, amongall of them, the one determined by a vertical straight line passing through thecenter of symmetry is the unique minimizing bisection, by applying Propo-sition 3.4 and Theorem 3.6. We point out that certain variations of this setprovide examples with infinite standard bisections, being none of them mini-mizing. p Figure 13.
Minimizing bisection for a cap bodyFinally, for the centrally symmetric planar convex body from Figure 14,obtained by cutting symmetrically a given ellipse, we have that the associatedstandard bisection is not minimizing. By using Theorem 3.6, it follows that aminimizing bisection is the one shown in Figure 14. p Figure 14.
Minimizing bisection for this centrally symmetricplanar convex body 6.
Some remarks
We finish these notes with some comments related with our optimizationproblem.
INIMIZING BISECTIONS FOR THE MAXIMUM RELATIVE DIAMETER 15
Optimal set.
Another interesting question for this problem is searchingfor the optimal sets , that is, the centrally symmetric planar convex bodies ofunit area with the minimum possible value for the maximum relative diameterfunctional. The unit-area condition here is required just as a normalizationfor the sets of our class. In this setting, the optimal set is unique and has beenobtained in [11, Example 2.3 and Th. 5]: it consists of the intersection of acertain strip (delimited by two parallel lines) and a symmetric lens.6.2.
Dual problems.
There are some dual optimization problems to the onediscussed in this paper, but they are worthless since their solutions are trivial.For instance, if we are interested in the bisections attaining the maximum possible value for d M , it is clear that we can consider a bisection determinedby a diameter segment of the set (and so, the diameter will be such maximumvalue). In fact, this will happen for any bisection with a subset containing twopoints whose distance equals the diameter of the set.On the other hand, we can consider the minimum relative diameter func-tional, defined as d m ( P ) = min { D ( C ) , D ( C ) } , where P is a bisection with subsets C , C . This functional has been alreadystudied in some previous works, see [6, 7]. It is easy to check that d m tends tozero for bisections with one of its associated subsets being reduced to a point,and that its maximum value will be attained again by a bisection given by adiameter segment.6.3. Relation with the Borsuk number.
For a given C ∈ C , the opti-mization problem for the maximum relative diameter functional treated inthis paper is meaningless when that functional is constant over all the bisec-tions of C (in that case, all the bisections can be seen as minimizing). Thissituation only happens when C is a circle, and it is equivalent to the follow-ing property: the unique centrally symmetric planar convex body with Borsuknumber equal to three is the circle (see [10, 3] and references therein for detailson this question). Acknowledgements.
The first author is partially supported by the projectMTM2013-48371-C2-1-P (Ministerio de Econom´ıa e Innovaci´on), and by Juntade Andaluc´ıa grant FQM-325 (Consejer´ıa de Econom´ıa, Innovaci´on, Cienciay Empleo). The second author is partially supported by MINECO/FEDERproject MTM2015-65430-P and “Programa de Ayudas a Grupos de Excelenciade la Regi´on de Murcia”, Fundaci´on S´eneca, 19901/GERM/15.
References [1] L. Bieberbach, ¨Uber eine Extremaleigenschaft des Kreises , Jber. Deutsch. Math.-Vereinig (1915), 247–250.[2] K. Borsuk, Drei S¨atze ¨uber die n-dimensionale euklidische Sph¨are , Fund. Math. (1933), 177–190.[3] A. Ca˜nete, Borsuk number and the diameter graph for planar convex bodies , preprint,2017. [4] A. Ca˜nete, C. Miori, S. Segura Gomis,
Trisections of a 3-rotationally symmetricplanar convex body miniminizing the maximum relative diameter , J. Math. Anal. Appl. (2014), no. 2, 1030–1046.[5] A. Ca˜nete, U. Schnell, S. Segura Gomis,
Subdivisions of rotationally symmetric planarconvex bodies minimizing the maximum relative diameter , J. Math. Anal. Appl. (2016), no. 1, 718–734.[6] A. Cerd´an, C. Miori, S. Segura Gomis,
Relative isodiametric inequalities , Beitr¨ageAlgebra Geom. (2004), no. 2, 595–605.[7] A. Cerd´an, U. Schnell, S. Segura Gomis, On relative geometric inequalities , Math.Inequal. Appl. (2004), no. 1, 135–148.[8] H. T. Croft, K. J. Falconer, R. K. Guy, Unsolved problems in Geometry , Springer-Verlag, Berlin, 1991.[9] H. Jung,
Ueber die kleinste Kugel, die eine r¨aumliche Figur einschliesst , J. ReineAngew. Math. (1901), 241–257.[10] D. Kolodziejczyk, Some remarks on the Borsuk conjecture,
Comment. Math. PraceMat. (1988), no. 1, 77–86.[11] C. Miori, C. Peri, S. Segura Gomis, On fencing problems , J. Math. Anal. Appl. (2004), no. 2, 464–476.[12] R. Osserman,
The isoperimetric inequality , Bull. Amer. Math. Soc. (1978), no. 6,1182-1238.[13] E. Schmidt. ¨Uber eine neue Methode zur Behandlung einer Klasse isoperimetrischerAufgaben im Grossen . Math. Z., 47 (1942), 489-642.[14] P. R. Scott, P. W. Awyong, Inequalities for convex sets , J. Inequal. Pure Appl. Math. (2000), no. 1, article 6 (electronic). Departamento de Matem´atica Aplicada I, Universidad de Sevilla
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