The geometry of homothetic covering and illumination
TThe geometry of homothetic covering and illumination
K´aroly Bezdek and Muhammad A. Khan
Abstract
At a first glance, the problem of illuminating the boundary of a convex body by external light sourcesand the problem of covering a convex body by its smaller positive homothetic copies appear to be quitedifferent. They are in fact two sides of the same coin and give rise to one of the important longstandingopen problems in discrete geometry, namely, the Illumination Conjecture. In this paper, we survey theactivity in the areas of discrete geometry, computational geometry and geometric analysis motivated bythis conjecture. Special care is taken to include the recent advances that are not covered by the existingsurveys. We also include some of our recent results related to these problems and describe two newapproaches – one conventional and the other computer-assisted – to make progress on the illuminationproblem. Some open problems and conjectures are also presented.
Keywords and phrases:
Illumination number, Illumination Conjecture, Covering Conjecture, Separa-tion Conjecture, X-ray number, X-ray Conjecture, illumination parameter, covering parameter, coveringindex, cylindrical covering parameters, (cid:15) -net of convex bodies.
MSC (2010): “ . . . N k bezeichne die kleinste nat¨urliche Zahl, f¨ur welche die nachfolgende Aussage richtig ist:Ist A ein eigentlicher konvexer K¨orper des k -dimensionalen euklidischen Raumes, so gibt es n mit A translations-gleiche K¨orper A i mit n ≤ N k derart, dass jeder Punkt von A ein innerer Punkt der Vere-inigungsmenge (cid:83) i A i ist, . . . Welchen Wert hat N k f¨ur k ≥ ?” [53] The above statement roughly translates to “Let N k denote the smallest natural number such that any k -dimensional convex body can be covered by the interior of a union of at the most N k of its translates. Whatis N k for k ≥ . However, he was not the first one to studythis particular problem. In fact, its earliest occurrence can be traced back to Levi’s 1955 paper [64], whoformulated and settled the 2-dimensional case of the problem. Later in 1960, the question was restated byGohberg and Markus [51] in terms of covering by homothetic copies. The equivalence of both formulationsis relatively easy to check and details appear in Section 34 of [36]. Conjecture 1.1 ( Covering Conjecture ) . We can cover any d -dimensional convex body by d or fewer ofits smaller positive homothetic copies in Euclidean d -space, d ≥ . Furthermore, d homothetic copies arerequired only if the body is an affine d -cube. The same conjecture has also been referred to in the literature as the Levi–Hadwiger Conjecture,Gohberg–Markus Covering Conjecture and Hadwiger Covering Conjecture. The condition d ≥ The Hadwiger conjecture in graph theory is, in the words of Bollob´as, Catalin and Erd˝os [29], “ one of the deepest unsolvedproblems in graph theory ”. Hadwiger even edited a column on unsolved problems in the journal
Elemente der Mathematik .On the occasion of Hadwiger’s 60th birthday, Victor Klee dedicated the first article in the Research Problems section of the
American Mathematical Monthly to Hadwiger’s work on promoting research problems [49, pp. 389–390]. Apparently, Gohberg and Markus worked on the problem independently without knowing about the work of Levi andHadwiger [33]. a r X i v : . [ m a t h . M G ] J u l et us make things formal. A d -dimensional convex body K is a compact convex subset of the Euclidean d -space, E d with nonempty interior. Let o denote the origin of E d . Then K is said to be o -symmetric if K = − K and centrally symmetric if some translate of K is o -symmetric. Since the quantities studied inthis paper are invariant under affine transformations, we use the terms o -symmetric and centrally symmetricinterchangeably. A homothety is an affine transformation of E d of the form x (cid:55)→ t + λ x , where t ∈ E d and λ is a non-zero real number. The image t + λ K of a convex body K under a homothety is said to be its homothetic copy (or simply a homothet ). A homothetic copy is positive if λ > negative otherwise.Furthermore, a homothetic copy with 0 < λ < K ⊆ E d , there exist t i ∈ E d and0 < λ i <
1, for i = 1 , . . . , d , such that K ⊆ d (cid:91) i =1 ( t i + λ i K ) . (1)Figure 1: A cube can be covered by 8 smaller positive homothets and no fewer.A light source at a point p outside a convex body K ⊂ E d , illuminates a point x on the boundary of K if the halfline originating from p and passing through x intersects the interior of K at a point not lyingbetween p and x . The set of points { p i : i = 1 , . . . , n } in the exterior of K is said to illuminate K if everyboundary point of K is illuminated by some p i . The illumination number I ( K ) of K is the smallest n forwhich K can be illuminated by n point light sources. p x (a) p p p (b) Figure 2: (a) Illuminating a boundary point x of K ⊂ E d by the point light source p ∈ E d \ K , (b) I ( K ) = 3.One can also consider illumination of K ⊂ E d by parallel beams of light. Let S d − be the unit spherecentered at the origin o of E d . We say that a point x on the boundary of K is illuminated in the direction v ∈ S d − if the halfline originating from x and with direction vector v intersects the interior of K .The former notion of illumination was introduced by Hadwiger [54], while the latter notion is due toBoltyanski [30]. It may not come as a surprise that the two concepts are equivalent in the sense that aconvex body K can be illuminated by n point sources if and only if it can be illuminated by n directions. Vladimir Boltyanski (also written Boltyansky, Boltyanskii and Boltjansky) is a prolific mathematician and recepient ofLenin Prize in science. He has authored more than 220 mathematical works including, remarkably, more than 50 books! v (a) (b) Figure 3: (a) Illuminating a boundary point x of K ⊂ E d by a direction v ∈ S d − , (b) I ( K ) = 3.However, it is less obvious that any covering of K by n smaller positive homothetic copies corresponds toilluminating K by n points (or directions) and vice versa (see [36] for details). Therefore, the followingIllumination Conjecture [30, 36, 54] of Hadwiger and Boltyanski is equivalent to the Covering Conjecture. Conjecture 1.2 ( Illumination Conjecture ) . The illumination number I ( K ) of any d -dimensional convexbody K , d ≥ , is at most d and I ( K ) = 2 d only if K is an affine d -cube. Figure 4: Vladimir Boltyanski (left, courtesy Annals of the Moscow University) and Hugo Hadwiger (right,courtesy Oberwolfach Photo Collection), two of the main proponents of the illumination problem.The conjecture also asserts that affine images of d -cubes are the only extremal bodies. The conjecturedbound of 2 d results from the 2 d vertices of an affine cube, each requiring a different light source to be illu-minated. In the sequel, we use the titles Covering Conjecture and Illumination Conjecture interchangeably,shifting between the covering and illumination paradigms as convenient.We have so far seen three equivalent formulations of the Illumination Conjecture. But there are more.In fact, it is perhaps an indication of the richness of this problem that renders it to be studied from manyangles, each with its own intuitive significance. We state one more equivalent form found independently byP. Soltan and V. Soltan [77], who formulated it for the o -symmetric case only and the first author [9, 10]. Conjecture 1.3 ( Separation Conjecture ) . Let K be an arbitrary convex body in E d , d ≥ , and o bean arbitrary interior point of K . Then there exist d hyperplanes of E d such that each intersection of K with a supporting hyperplane, called a face of K , can be strictly separated from o by at least one of the d hyperplanes. Furthermore, d hyperplanes are needed only if K is the convex hull of d linearly independentline segments which intersect at the common relative interior point o . Over the years, the illumination conjecture has inspired a vast body of research in convex and discretegeometry, computational geometry and geometric analysis. There exist some nice surveys on the topic suchas the papers [17, 66] and the corresponding chapters of the books [23, 36]. However, most of these are a bitdated. Moreover, we feel that the last few years have seen some interesting new ideas, such as the possibilityof a computer-assisted proof, that are not covered by any of the above-mentioned surveys. The aim of this3aper is to provide an accessible introduction to the geometry surrounding the Illumination Conjecture anda snapshot of the research motivated by it, with special emphasis on some of the recent developments. Atthe same time we describe some of our new results in this area.We organize the material as follows. Section 2 gives a brief overview on the progress of the IlluminationConjecture. In Section 3, we mention some important relatives of the illumination problem, while Section4 explores the known important quantitative versions of the problem including a new approach to makeprogress on the Illumination Conjecture based on the covering index of convex bodies (see Problem 3 andthe discussion following it in Section 4.2). Finally, in Section 5 we present Zong’s computer-assisted approach[88] for possibly resolving the Illumination Conjecture in low dimensions. E and E Despite its intuitive richness, the illumination conjecture has been notoriously difficult to crack even in thefirst nontrivial case of d = 3. The closest anything has come is the proof announced by Boltyanski [37]for the 3-dimensional case. Unfortunately, the proof turned out to have gaps that remain to date. Later,Boltyanski [38] modified his claim to the following. Theorem 2.1.
Let K be a convex body of E with md K = 2 . Then I ( K ) ≤ . Here md is a functional introduced by Boltyanski in [31] and defined as follows for any d -dimensionalconvex body: Let K ⊆ E d be a convex body. Then md( K ) is the greatest integer m for which there exist m + 1 regular boundary points of K such that the outward unit normals v , . . . , v m of K at these points areminimally dependent, i.e., they are the vertices of an m -dimensional simplex that contains the origin in itsrelative interior. So far the best upper bound on illumination number in three dimensions is due to Papadoperakis [70].
Theorem 2.2.
The illumination number of any convex body in E is at most 16. However, there are partial results that establish the validity of the conjecture for some large classes ofconvex bodies. Often these classes of convex bodies have some underlying symmetry. Here we list some suchresults. A convex polyhedron P is said to have affine symmetry if the affine symmetry group of P consistsof the identity and at least one other affinity of E . The first author obtained the following result [9]. Theorem 2.3. If P is a convex polyhedron of E with affine symmetry, then the illumination number of P is at most 8. Recall that a convex body K is said to be centrally symmetric if it has a point of symmetry. Furthermore,a body K is symmetric about a plane p if a reflection across that plane leaves K unchanged. Lassak [59]proved that under the assumption of central symmetry, the illumination conjecture holds in three dimensions. Theorem 2.4. If K is a centrally symmetric convex body in E , then I ( K ) ≤ . Dekster [45] extended Theorem 2.3 from polyhedra to convex bodies with plane symmetry.
Theorem 2.5. If K is a convex body symmetric about a plane in E , then I ( K ) ≤ . It turns out that for 3-dimensional bodies of constant width – that is bodies whose width, measured bythe distance between two opposite parallel hyperplanes touching its boundary, is the same regardless of thedirection of those two parallel planes – we get an even better bound.
Theorem 2.6.
The illumination number of any convex body of constant width in E is at most 6. In fact, it is proved in [31] that md( K ) = him( K ) holds for any convex body K of E d and therefore one can regardTheorem 2.1 as an immediate corollary of Theorem 2.16 for d = 3 in Section 2.2. Conjecture 2.7.
The illumination number of any convex body of constant width in E is exactly 4. The above conjecture, if true, would provide a new proof of Borsuk’s conjecture [42] in dimension three,which states that any set of unit diameter in E can be partitioned into at most four subsets of diameter lessthan one. We remark that although it is false in general [56], Borsuk’s conjecture has a long and interestinghistory of its own and the reader can look up [32, 36, 52] for detailed discussions.Now let us consider the state of the Illumination Conjecture in E . It is well known that neighbourly d -polytopes have the maximum number of facets among d -polytopes with a fixed number of vertices (formore details on this see for example, [28]). Thus, it is natural to investigate the Separation Conjecture forneighbourly d -polytopes (see also Theorem 2.18). Since interesting neighbourly d -polytopes exist only in E d for d ≥
4, it is particularly natural to first restrict our attention to neighbourly 4-polytopes. Startingfrom a cyclic 4-polytope, the sewing procedure of Shemer (for details see [28]) produces an infinite familyof neighbourly 4-polytopes each of which is obtained from the previous one by adding one new vertex ina suitable way. Neighbourly 4-polytopes obtained from a cyclic 4-polytope by a sequence of sewings arecalled totally-sewn . The main result of the very recent paper [28] of Bisztriczky and Fodor is a proof of theSeparation Conjecture for totally-sewn neighbourly 4-polytopes.
Theorem 2.8.
Let P be an arbitrary totally-sewn neighbourly -polytope in E , and o be an arbitrary interiorpoint of P . Then there exist hyperplanes of E such that each face of P , can be strictly separated from o by at least one of the hyperplanes. However, Bisztriczky [27] conjectures the following stronger result.
Conjecture 2.9.
Let P be an arbitrary totally-sewn neighbourly -polytope in E , and o be an arbitraryinterior point of P . Then there exist hyperplanes of E such that each face of P , can be strictly separatedfrom o by at least one of the hyperplanes. Before we state results on the illumination number of convex bodies in E d , we take a little detour. We needRogers’ estimate [71] of the infimum θ ( K ) of the covering density of E d by translates of the convex body K ,namely, for d ≥ θ ( K ) ≤ d (ln d + ln ln d + 5)and the Rogers–Shephard inequality [72]vol d ( K − K ) ≤ (cid:18) dd (cid:19) vol d ( K )on the d -dimensional volume vol d ( · ) of the difference body K − K of K .It was rather a coincidence, at least from the point of view of the Illumination Conjecture, when in 1964Erd˝os and Rogers [47] proved the following theorem. In order to state their theorem in a proper form weneed to introduce the following notion. If we are given a covering of a space by a system of sets, the starnumber of the covering is the supremum, over sets of the system, of the cardinals of the numbers of setsof the system meeting a set of the system (see [47]). On the one hand, the standard Lebesgue brick-layingconstruction provides an example, for each positive integer d , of a lattice covering of E d by closed cubes withstar number 2 d +1 −
1. On the other hand, Theorem 1 of [47] states that the star number of a lattice coveringof E d by translates of a centrally symmetric convex body is always at least 2 d +1 −
1. However, from ourpoint of view, the main result of [47] is the one under Theorem 2 which (combined with some observationsfrom [46] and the Rogers–Shephard inequality [71]) reads as follows. The bound on θ ( K ) has been improved to θ ( K ) ≤ d ln d + d ln ln d + d + o ( d ) by G. Fejes T´oth [48]. heorem 2.10. Let K be a convex body in the d -dimensional Euclidean space E d , d ≥ . Then there existsa covering of E d by translates of K with star number at most vol d ( K − K )vol d ( K ) ( d ln d + d ln ln d + 5 d + 1) ≤ (cid:18) dd (cid:19) ( d ln d + d ln ln d + 5 d + 1) . Moreover, for sufficiently large d , d can be replaced by d . The periodic and probabilistic construction on which Theorem 2.10 is based gives also the following.
Corollary 2.11. If K is an arbitrary convex body in E d , d ≥ , then I ( K ) ≤ vol d ( K − K )vol d ( K ) d (ln d + ln ln d + 5) ≤ (cid:18) dd (cid:19) d (ln d + ln ln d + 5) = O (4 d √ d ln d ) . (2) Moreover, for sufficiently large d , d can be replaced by d . Note that the bound given in Corollary 2.11 can also be obtained from the more general result of Rogersand Zong [73], which states that for d -dimensional convex bodies K and L , d ≥
2, one can cover K by N ( K , L ) translates of L such that N ( K , L ) ≤ vol d ( K − L )vol d ( L ) θ ( L ) . For the sake of completeness we also mention the inequality I ( K ) ≤ ( d + 1) d d − − ( d − d − d − due to Lassak [61], which is valid for an arbitrary convex body K in E d , d ≥
2, and is (somewhat) betterthan the estimate of Corollary 2.11 for some small values of d .Since, for a centrally symmetric convex body K , vol( K − K )vol d ( K ) = 2 d , we have the following improved upperbound on the illumination number of such convex bodies. Corollary 2.12. If K is a centrally symmetric convex body in E d , d ≥ , then I ( K ) ≤ vol d ( K − K )vol d ( K ) d (ln d + ln ln d + 5) = 2 d d (ln d + ln ln d + 5) = O (2 d d ln d ) . (3)The above upper bound is fairly close to the conjectured value of 2 d . However, most convex bodies arefar from being symmetric and so, in general, one may wonder whether the Illumination Conjecture is trueat all, especially for large d . Thus, it was important progress when Schramm [75] managed to prove theIllumination Conjecture for all convex bodies of constant width in all dimensions at least 16. In fact, heproved the following inequality. Theorem 2.13. If W is an arbitrary convex body of constant width in E d , d ≥ , then I ( W ) ≤ d √ d (4 + ln d ) (cid:18) (cid:19) d . By taking a closer look of Schramm’s elegant paper [75] and making the necessary modifications, thefirst author [24] somewhat improved the upper bound of Theorem 2.13, but more importantly he succeededin extending that estimate to the following family of convex bodies (called the family of fat spindle convexbodies ) that is much larger than the family of convex bodies of constant width. Thus, we have the followinggeneralization of Theorem 2.13 proved in [24]. N ( K , L ) is called the covering number of K by L . heorem 2.14. Let X ⊂ E d , d ≥ be an arbitrary compact set with diam( X ) ≤ and let B [ X ] be theintersection of the closed d -dimensional unit balls centered at the points of X . Then I ( B [ X ]) < (cid:16) π (cid:17) d (3 + ln d ) (cid:18) (cid:19) d < d (4 + ln d ) (cid:18) (cid:19) d . On the one hand, 4 (cid:0) π (cid:1) d (3 + ln d ) (cid:0) (cid:1) d < d for all d ≥
15. (Moreover, for every (cid:15) > d issufficiently large, then I ( B [ X ]) < (cid:0) √ . (cid:15) (cid:1) d = (1 . . . . + (cid:15) ) d .) On the other hand, based on the elegantconstruction of Kahn and Kalai [56], it is known (see [1]), that if d is sufficiently large, then there exists afinite subset X (cid:48)(cid:48) of { , } d in E d such that any partition of X (cid:48)(cid:48) into parts of smaller diameter requires morethan (1 . √ d parts. Let X (cid:48) be the (positive) homothetic copy of X (cid:48)(cid:48) having unit diameter and let X be the(not necessarily unique) convex body of constant width one containing X (cid:48) . Then it follows via standardarguments that I ( B [ X ]) > (1 . √ d with X = B [ X ].Recall that a convex polytope is called a belt polytope if to each side of any of its 2-faces there existsa parallel (opposite) side on the same 2-face. This class of polytopes is wider than the class of zonotopes.Moreover, it is easy to see that any convex body of E d can be represented as a limit of a covergent sequenceof belt polytopes with respect to the Hausdorff metric in E d . The following theorem on belt polytopeswas proved by Martini in [65]. The result that it extends to the class of convex bodies, called belt bodies(including zonoids), is due to Boltyanski [34, 35, 36]. (See also [39] for a somewhat sharper result on theillumination numbers of belt bodies.) Theorem 2.15.
Let P be an arbitrary d -dimensional belt polytope (resp., belt body) different from a paral-lelotope in E d , d ≥ . Then I ( P ) ≤ · d − . Now, let K be an arbitrary convex body in E d and let T ( K ) be the family of all translates of K in E d .The Helly dimension him( K ) of K ([76]) is the smallest integer h such that for any finite family F ⊆ T ( K )with cardinality greater than h + 1 the following assertion holds: if every h + 1 members of F have a pointin common, then all the members of F have a point in common. As is well known 1 ≤ him( K ) ≤ d . Usingthis notion Boltyanski [38] gave a proof of the following theorem. Theorem 2.16.
Let K be a convex body with him( K ) = 2 in E d , d ≥ . Then I ( K ) ≤ d − d − . In fact, in [38] Boltyanski conjectures the following more general inequality.
Conjecture 2.17.
Let K be a convex body with him( K ) = h > in E d , d ≥ . Then I ( K ) ≤ d − d − h . The first author and Bisztriczky gave a proof of the Illumination Conjecture for the class of dual cyclicpolytopes in [15]. Their upper bound for the illumination numbers of dual cyclic polytopes has been improvedby Talata in [79]. So, we have the following statement.
Theorem 2.18.
The illumination number of any d -dimensional dual cyclic polytope is at most ( d +1) , forall d ≥ . Let K be a convex body in E d , d ≥
2. The following definitions were introduced by the first named authorin [14] (see also [9] that introduced the concept of the first definition below).7et L ⊂ E d \ K be an affine subspace of dimension l , 0 ≤ l ≤ d −
1. Then L illuminates the boundarypoint q of K if there exists a point p of L that illuminates q on the boundary of K . Moreover, we saythat the affine subspaces L , L , . . . , L n of dimension l with L i ⊂ E d \ K , ≤ i ≤ n illuminate K if everyboundary point of K is illuminated by at least one of the affine subspaces L , L , . . . , L n . Finally, let I l ( K )be the smallest positive integer n for which there exist n affine subspaces of dimension l say, L , L , . . . , L n such that L i ⊂ E d \ K for all 1 ≤ i ≤ n and L , L , . . . , L n illuminate K . Then I l ( K ) is called the l -dimensional illumination number of K and the sequence I ( K ) , I ( K ) , . . . , I d − ( K ) , I d − ( K ) is called the successive illumination numbers of K . Obviously, I ( K ) = I ( K ) ≥ I ( K ) ≥ · · · ≥ I d − ( K ) ≥ I d − ( K ) = 2.Recall that S d − denotes the unit sphere centered at the origin of E d . Let HS l ⊂ S d − be an l -dimensionalopen great-hemisphere of S d − , where 0 ≤ l ≤ d −
1. Then HS l illuminates the boundary point q of K ifthere exists a unit vector v ∈ HS l that illuminates q , in other words, for which it is true that the halflineemanating from q and having direction vector v intersects the interior of K . Moreover, we say that the l -dimensional open great-hemispheres HS l , HS l , . . . , HS ln of S d − illuminate K if each boundary point of K is illuminated by at least one of the open great-hemispheres HS l , HS l , . . . , HS ln . Finally, let I (cid:48) l ( K )be the smallest number of l -dimensional open great-hemispheres of S d − that illuminate K . Obviously, I (cid:48) ( K ) ≥ I (cid:48) ( K ) ≥ · · · ≥ I (cid:48) d − ( K ) ≥ I (cid:48) d − ( K ) = 2.Let L ⊂ E d be a linear subspace of dimension l , 0 ≤ l ≤ d − E d . The l -codimensional circumscribedcylinder of K generated by L is the union of translates of L that have a nonempty intersection with K .Then let C l ( K ) be the smallest number of translates of the interiors of some l -codimensional circumscribedcylinders of K the union of which contains K . Obviously, C ( K ) ≥ C ( K ) ≥ · · · ≥ C d − ( K ) ≥ C d − ( K ) = 2.The following theorem, which was proved in [14], collects the basic information known about the quantitiesjust introduced. Theorem 3.1.
Let K be an arbitrary convex body of E d . Then(i) I l ( K ) = I (cid:48) l ( K ) = C l ( K ) , for all ≤ l ≤ d − .(ii) (cid:100) d +1 l +1 (cid:101) ≤ I l ( K ) , for all ≤ l ≤ d − , with equality for any smooth K .(iii) I d − ( K ) = 2 , for all d ≥ . The Generalized Illumination Conjecture was phrased by the first named author in [14] as follows.
Conjecture 3.2 ( Generalized Illumination Conjecture).
Let K be an arbitrary convex body and C d be a d -dimensional affine cube in E d . Then I l ( K ) ≤ I l ( C d ) holds for all ≤ l ≤ d − . The above conjecture was proved for zonotopes and zonoids in [14]. The results of parts (i) and (ii) ofthe next theorem are taken from [14], where they were proved for zonotopes (resp., zonoids). However, inthe light of the more recent works in [35] and [39] these results extend to the class of belt polytopes (resp.,belt bodies) in a rather straightforward way so we present them in that form. The lower bound of part (iii)was proved in [14] and the upper bound of part (iii) is the major result of [57]. Finally, part (iv) was provedin [13].
Theorem 3.3.
Let M be a belt polytope (resp., belt body) and C d be a d -dimensional affine cube in E d .Then(i) I l ( M ) ≤ I l ( C d ) holds for all ≤ l ≤ d − .(ii) I (cid:98) d (cid:99) ( M ) = · · · = I d − ( M ) = 2 .(iii) d (cid:80) li =0 ( di ) ≤ I l ( C d ) ≤ K ( d, l ) , where K ( d, l ) denotes the minimum cardinality of binary codes of length d with covering radius l , ≤ l ≤ d − .(iv) I ( C d ) = d d +1 , provided that d + 1 = 2 m . .2 ‘X-raying’ the problemx L (a) (b) Figure 5: (a) X-raying a boundary point x of K along a line L , (b) X ( K ) = 2.In 1972, the X-ray number of convex bodies was introduced by P. Soltan as follows (see [66]). Let K be a convex body of E d , d ≥
2, and L ⊂ E d be a line through the origin of E d . We say that the boundarypoint x ∈ K is X-rayed along L if the line parallel to L passing through x intersects the interior of K . TheX-ray number X ( K ) of K is the smallest number of lines such that every boundary point of K is X-rayedalong at least one of these lines. Clearly, X ( K ) ≥ d . Moreover, it is easy to see that this bound is attainedby any smooth convex body. On the other hand, if C d is a d -dimensional (affine) cube and F is one of its( d − C d \ F , the convex hull of the set of vertices of C d thatdo not belong to F , is 3 · d − .In 1994, the first author and Zamfirescu [18] published the following conjecture. Conjecture 3.4 ( X-ray Conjecture).
The X-ray number of any convex body in E d is at most · d − . The X-ray Conjecture is proved only in the plane and it is open in higher dimensions. Here we note thatthe inequalities X ( K ) ≤ I ( K ) ≤ X ( K )hold for any convex body K ⊂ E d . In other words, any proper progress on the X-ray Conjecture wouldimply progress on the Illumination Conjecture and vice versa. We also note that a natural way to prove theX-ray Conjecture would be to show that any convex body K ⊂ E d can be illuminated by 3 · d − pairs ofpairwise opposite directions.The main results of [22] on the X-ray number can be summarized as follows. In order to state themproperly we need to recall two basic notions. Let K be a convex body in E d and let F be a face of K .The Gauss image ν ( F ) of the face F is the set of all points (i.e., unit vectors) u of the ( d − S d − ⊂ E d centered at the origin o of E d for which the supporting hyperplane of K with outer normalvector u contains F . It is easy to see that the Gauss images of distinct faces of K have disjoint relativeinteriors in S d − and ν ( F ) is compact and spherically convex for any face F . Let C ⊂ S d − be a set offinitely many points. Then the covering radius of C is the smallest positive real number r with the propertythat the family of spherical balls of radii r centered at the points of C covers S d − . Theorem 3.5.
Let K ⊂ E d , d ≥ , be a convex body and let r be a positive real number with the property thatthe Gauss image ν ( F ) of any face F of K can be covered by a spherical ball of radius r in S d − . Moreover,assume that there exist m pairwise antipodal points of S d − with covering radius R satisfying the inequality r + R ≤ π . Then X ( K ) ≤ m . In particular, if there are m pairwise antipodal points on S d − with coveringradius R satisfying the inequality R ≤ π/ − r d − , where r d − = arccos (cid:113) d +12 d is the circumradius of a regular ( d − -dimensional spherical simplex of edge length π/ , then X ( W ) ≤ m holds for any convex body W ofconstant width in E d . Theorem 3.6. If W is an arbitrary convex body of constant width in E , then X ( W ) = 3 . If W is anyconvex body of constant width in E , then ≤ X ( W ) ≤ . Moreover, if W is a convex body of constantwidth in E d with d = 5 , , then d ≤ X ( W ) ≤ d − . orollary 3.7. If W is an arbitrary convex body of constant width in E , then ≤ I ( W ) ≤ . If W isany convex body of constant width in E , then ≤ I ( W ) ≤ . Moreover, if W is a convex body of constantwidth in E d with d = 5 , , then d + 1 ≤ I ( W ) ≤ d . It would be interesting to extend the method described in the paper [22] for the next few dimensions(more exactly, for the dimensions 7 ≤ d ≤
14) in particular, because in these dimensions neither the X-rayConjecture nor the Illumination Conjecture is known to hold for convex bodies of constant width.From the proof of Theorem 2.4 it follows in a straightforward way that if K is a centrally symmetricconvex body in E , then X ( K ) ≤
4. On the other hand, very recently Trelford [80] proved the followingrelated result.
Theorem 3.8. If K is a convex body symmetric about a plane in E , then X ( K ) ≤ . t -covering and t -illumination numbers In Section 1, we found that the least number of smaller positive homothets of a convex body K required tocover it equals the minimum number of translates of the interior of K needed to cover K . Is this numberalso equal to the the minimum number t ( K ) of translates of K that are different from K and are needed tocover K ?Despite being a very natural question, the problem of economical translative coverings have not attractedmuch attention. To our knowledge, the first systematic study of these was carried out quite recently by Las-sak, Martini and Spirova [63] who called them t -coverings and also introduced the corresponding illuminationconcept, called t -illumination, as follows: A boundary point x of a convex body K of E d is t -illuminated by a direction v ∈ S d − if there exists a different point y ∈ K such that the vector y − x has the samedirection as v (i.e., y − x = λ v , for some λ > i ( K ) of directions needed to t -illuminate the entire boundary of K is called its t -illumination number. The connection between t -coveringand t -illumination is summarized in the next result [63]. Note that a convex body K is said to be strictlyconvex if for any two points of K the open line segment connecting them belongs to the interior of K . Theorem 3.9. (i) If K is a planar convex body, then i ( K ) = t ( K ) .(ii) If K is a d -dimensional strictly convex body, d ≥ , then i ( K ) = t ( K ) .(iii) If K is a d -dimensional convex body, d ≥ , then i ( K ) ≤ t ( K ) , where the equality does not hold ingeneral. Clearly, t ( K ) ≤ I ( K ). In the same paper [63], the following results were obtained about the relationshipbetween I ( · ) and t ( · ). Theorem 3.10. (i) If K is a planar convex body, then t ( K ) = I ( K ) if and only if K contains no parallel boundary segments.(i) If K is a strictly convex body of E d , d ≥ , then t ( K ) = I ( K ) .(iii) If K is a convex body of E d , d ≥ that does not have parallel boundary segments, then t ( K ) = I ( K ) . However, in general the following remains unanswered [63].
Problem 1.
Characterize the convex bodies K for which t ( K ) = I ( K ) . In [67], the notion of t -illumination was refined into t -central illumination and strict t -illumination and thecorresponding illumination numbers were defined. The paper also introduced metric versions of the classical, t -central and strict t -illumination numbers and investigated their properties at length. The interested readeris referred to [67] for details. 10 .3.2 Blocking numbers The blocking number β ( K ) [87] of a convex body K is defined as the minimum number of nonoverlappingtranslates of K that can be brought into contact with the boundary of K so as to block any other translate of K from touching K . Since β ( K ) = β ( K − K ) and K − K is o -symmetric, it suffices to consider the blockingnumbers of o -symmetric convex bodies only.For any o -symmetric convex body K , the relation I ( K ) ≤ β ( K ) holds [87], while no such relationshipexists for general convex bodies. Zong [87] conjectured the following. Conjecture 3.11.
For any d -dimensional convex body K , d ≤ β ( K ) ≤ d , and β ( K ) = 2 d if and only if K is a d -dimensional cube. If true, Zong’s Conjecture would imply the Illumination Conjecture for o -symmetric convex bodies. Someof the known values of the blocking number include β ( K ) = 2 d , if K is a d -dimensional cube; β ( K ) = 6, if K is a 3-dimensional ball; and β ( K ) = 9, if K is a 4-dimensional ball [44]. Some other values and estimatesare obtained in [86].Several generalizations of the blocking number have been proposed. The smallest number of non-overlapping translates of K such that the interior of K is disjoint from the interiors of the translates and theycan block any other translate from touching K is denoted by β ( K ); the smallest number of translates all ofwhich touch K at its boundary such that they can block any other translate from touching K is denoted by β ( K ); whereas, β ( K ) denotes the smallest number of translates all of which are non-overlapping with K such that they can block any other translate from touching K [85]. If in the original definition of blockingnumber, translates are replaced by homothets with homothety ratio α > generalized blockingnumber β α ( K ) [40], and if we allow the homothets to overlap, we get the generalized α -blocking number β α ( K ) [84].Recently, Wu [84] showed that if K and L are o -symmetric convex bodies that are sufficiently close toeach other in the Banach–Mazur sense then there exists α > K ) such that I ( K ) ≤ β α ( L ) . This gives a series of upper bounds on the illumination number of symmetric convex bodies and a possibleway to circumvent the lack of lower semicontinuity of I ( · ) (see Section 4.1 for a discussion of the continuityof the illumination number). Nasz´odi [68] introduced the fractional illumination number and Arstein-Avidan with Raz [3] and with Slomka[4] introduced weighted covering numbers. Both formalisms can be used to study a fractional analogue ofthe illumination problem. In fact, the Fractional Illumination Conjecture for o -symmetric convex bodies wasproved in [68], while the case of equality was characterized in [4]. We omit the details as it would lead to alengthy diversion from the main subject matter. It can be seen that in the definition of illumination number I ( K ), the distance of light sources from K playsno role whatsoever. Starting with a relatively small number of light sources, it makes sense to quantify how See relation (6) and the discussion preceding it in Section 4.1 for an introduction to the Banach–Mazur distance of convexbodies. Note that Wu uses the Hausdorff distance between convex bodies to state his result. However, it can be shown that K and L are close to each other in the Banach–Mazur sense if and only if there exist affine images of them that are close in theHausdorff sense. Since the illumination and blocking numbers are affine invariants, we can restate Wu’s results in the languageof Banach–Mazur distance. K in order to illuminate it. This is the idea behind the illumination parameter asdefined by the first author [11].Let K be an o -symmetric convex body. Then the norm of x ∈ E d generated by K is defined as (cid:107) x (cid:107) K = inf { λ > x ∈ λ K } and provides a good estimate of how far a point x is from K .The illumination parameter ill( K ) of an o -symmetric convex body K estimates how well K can beilluminated by relatively few point sources lying as close to K on average as possible.ill( K ) = inf (cid:40)(cid:88) i (cid:107) p i (cid:107) K : { p i } illuminates K , p i ∈ E d (cid:41) , Clearly, I ( K ) ≤ ill( K ) holds for any o -symmetric convex body K . In the papers [16, 58], the illuminationparameters of o -symmetric Platonic solids have been determined. In [11] a tight upper bound was obtainedfor the illumination parameter of planar o -symmetric convex bodies. Theorem 4.1. If K is an o -symmetric planar convex body, then ill( K ) ≤ with equality for any affineregular convex hexagon. The corresponding problem in dimension 3 and higher is wide open. The following conjecture is due toKiss and de Wet [58].
Conjecture 4.2.
The illumination parameter of any o -symmetric 3-dimensional convex body is at most 12. However, for smooth o -symmetric convex bodies in any dimension d ≥
2, the first named author andLitvak [20] found an upper bound, which was later improved to the following asymptotically sharp boundby Gluskin and Litvak [50].
Theorem 4.3.
For any smooth o -symmetric d -dimensional convex body K , ill( K ) ≤ d / . Translating the above quantification ideas from illumination into the setting of covering, Swanepoel [78]introduced the covering parameter of a convex body as follows. C ( K ) = inf (cid:40)(cid:88) i (1 − λ i ) − : K ⊆ (cid:91) i ( λ i K + t i ) , < λ i < , t i ∈ E d (cid:41) . Thus large homothets are penalized in the same way as the far off light sources are penalized in thedefinition of illumination parameter. Note that here K need not be o -symmetric. In the same paper,Swanepoel obtained the following Rogers-type upper bounds on C ( K ) when d ≥ Theorem 4.4. C ( K ) < e d d ( d + 1)(ln d + ln ln d + 5) = O (2 d d ln d ) , if K is o -symmetric ,e (cid:18) dd (cid:19) d ( d + 1)(ln d + ln ln d + 5) = O (4 d d / ln d ) , otherwise . (4)He further showed that if K is o -symmetric, thenill( K ) ≤ C ( K ) , (5)and therefore, ill( K ) = O (2 d d ln d ).Based on the above results, it is natural to study the following quantitative analogue of the illuminationconjecture that was proposed by Swanepoel [78]. 12 onjecture 4.5 ( Quantitative Illumination Conjecture).
For any o -symmetric d -dimensional convexbody K , ill( K ) = O (2 d ) . Before proceeding further, we introduce some terminology and notations. Let us use K d and C d respec-tively to denote the set of all d -dimensional convex bodies and the set of all such bodies that are o -symmetric.In this section, we consider some of the important properties of the illumination number and the coveringparameter as functionals defined on K d and the illumination parameter as a functional on C d . The firstobservation is that the three quantities are affine invariants (as are several other quantities dealing with thecovering and illumination of convex bodies). That is, if A : E d → E d is an affine transformation and K isany d -dimensional convex body, then I ( K ) = I ( A ( K )), ill( K ) = ill( A ( K )) and C ( K ) = C ( A ( K )).Due to this affine invariance, whenever we refer to a convex body K , whatever we say about the coveringand illumination of K is true for all affine images of K . In the sequel, B d denotes a d -dimensional unit ball , C d a d -dimensional cube and (cid:96) a line segment (which is a convex body in K ) up to an affine transformation.The Banach–Mazur distance d BM provides a multiplicative metric on K d and is used to study thecontinuity properties of affine invariant functionals on K d . For K , L ∈ K d , it is given by d BM ( K , L ) = inf { δ ≥ L − b ⊆ T ( K − a ) ⊆ δ ( L − b ) , a ∈ K , b ∈ L } , (6)where the infimum is taken over all invertible linear operators T : E d → E d [74, Page 589].In the remainder of this paper, K d (resp., C d ) is considered as a metric space under the Banach–Mazurdistance. Since continuity of a functional can provide valuable insight into its behaviour, it is of considerableinterest to check the continuity of I ( · ), ill( · ) and C ( · ). Unfortunately, by Example 4.6, the first two quantitiesare known to be discontinuous, while nothing is known about the continuity of the third. Example 4.6 (Smoothed cubes and spiky balls) . In K d , consider a sequence ( C n ) n ∈ N of ‘smoothed’ d -dimensional cubes that approaches C d in the Banach–Mazur sense. Since the smoothed cubes are smoothconvex bodies, all the terms of the sequence have illumination number d + 1 . However, I ( C d ) = 2 d , whichshows that I ( · ) is not continuous.Recently, Nasz´odi [69] constructed a class of d -dimensional o -symmetric bodies, that he refers to as ‘spikyballs’. Pick N points x , . . . , x N independently and uniformly with respect to the Haar probability measureon the ( d − -dimensional unit sphere S d − centred at the origin o . Then a spiky ball corresponding to areal number D > is defined as K = conv (cid:18) {± x i : i = 1 , . . . , N } ∪ D B d (cid:19) . Straightaway we observe that K is o -symmetric and satisfies d BM ( K , B d ) < D . Nasz´odi showed that I ( K ) ≥ c d , where c > is a constant depending on d and D . Thus we have a sequence of spiky ballsapproaching B d in Banach–Mazur distance such that each spiky ball has an exponential illumination number.Since by Theorem 4.3, ill( B d ) = O ( d / ) and ill( K ) ≥ I ( K ) we see that ill( · ) is not continuous. We can state the following about the continuity of I ( · ) [36]. Theorem 4.7.
The functional I ( · ) is upper semicontinuous on E d , for all d ≥ . Despite the usefulness of the covering parameter, not much is known about it. For instance, we do notknow whether C ( · ) is lower or upper semicontinuous on K d and the only known exact value is C ( C d ) = 2 d +1 .Thus there is a need to propose a more refined quantitative version of homothetic covering for convex bodies.Section 4.2 describes how we address this need. Without loss of generality, we can assume B d to be a unit ball centred at the origin. In what follows, we use the symbol B d to denote a d -dimensional unit ball as well as its affine images called ellipsoids. One can turn the Banach–Mazur distance into an additive metric by applying ln( · ). However, we make no attempt to dothat. Again, the original statement of this result is in terms of Hausdorff distance. However, based on the discussion in footnote7, we can use the Banach–Mazur distance instead. .2 The covering index The concepts and results presented in this section appear in our recent paper [25]. As stated at the end of Sec-tion 4.1, the aim here is to come up with a more refined quantification of covering in terms of the covering in-dex with the Covering Conjecture as the eventual goal. The covering index of a convex body K in E d combinesthe notions of the covering parameter C ( K ) and the m -covering number γ m ( K ) under the unusual, but highlyuseful, constraint γ m ( K ) ≤ /
2, where γ m ( K ) = inf (cid:8) λ > K ⊆ (cid:83) mi =1 ( λ K + t i ) , t i ∈ E d , i = 1 , . . . , m (cid:9) isthe smallest positive homothety ratio needed to cover K by m positive homothets. (See Section 5 for adetailed discussion of γ m ( · ).) Definition 1 (Covering index) . Let K be a d -dimensional convex body. We write N λ ( K ) to denote thecovering number N ( K , λ K ) , for any < λ ≤ . We define the covering index of K as coin( K ) = inf (cid:26) m − γ m ( K ) : γ m ( K ) ≤ / , m ∈ N (cid:27) = inf (cid:26) N λ ( K )1 − λ : 0 < λ ≤ / (cid:27) . Intuitively, coin( K ) measures how K can be covered by a relatively small number of positive homothetsall corresponding to the same relatively small homothety ratio. The reader may be a bit surprised to seethe restriction γ m ( K ) ≤ /
2. In [25], it was observed that if we start with γ m ( K ) ≤ λ <
1, for some λ closeto 1 in the definition of coin( K ) and then decrease λ , the properties of the resulting quantity significantlychange when λ = 1 /
2, at which point we can say a lot about the continuity and maximum and minimumvalues of the quantity. It was also observed that decreasing λ further does not change these characteristics.Thus 1 / K ∈ C d , I ( K ) ≤ ill( K ) ≤ C ( K ) ≤ K ) , and in general for K ∈ K d , I ( K ) ≤ C ( K ) ≤ coin( K ) . The following result shows that a lot can be said about the Banach–Mazur continuity of coin( · ). Basedon this, coin( · ) seems to be the ‘nicest’ of all the functionals of covering and illumination of convex bodiesdiscussed here. Theorem 4.8.
Let d be any positive integer.(i) Define I K = { i : γ i ( K ) ≤ / } = { i : K ∈ K di } , for any d -dimensional convex body K . If I L ⊆ I K , forsome K , L ∈ K d , then coin( K ) ≤ d BM ( K , L ) − d BM ( K , L ) coin( L ) ≤ d BM ( K , L ) coin( L ) .(ii) The functional coin : K d → R is lower semicontinuous for all d .(iii) Define K d ∗ := (cid:8) K ∈ K d : γ m ( K ) (cid:54) = 1 / , m ∈ N (cid:9) . Then the functional coin : K d ∗ → R is continuous forall d . We now present some results showing that coin( · ) behaves very nicely with forming direct sums, Minkowskisums and cylinders of convex bodies, making it possible to compute the exact values and estimates of coin( · )for higher dimensional convex bodies from the covering indices of lower dimensional convex bodies. Theorem 4.9. (i) Let E d = L ⊕ · · · ⊕ L n be a decomposition of E d into the direct vector sum of its linear subspaces L i andlet K i ⊆ L i be convex bodies such that Γ = max { γ m i ( K i ) : 1 ≤ i ≤ n } . Then max ≤ i ≤ n { coin( K i ) } ≤ coin( K ⊕ · · · ⊕ K n ) ≤ inf λ ≤ (cid:81) ni =1 N λ ( K i )1 − λ ≤ (cid:81) ni =1 N Γ ( K i )1 − Γ < n (cid:89) i =1 coin( K i ) . (7) In fact, one can obtain ill( K ) ≤ coin( K ) by suitably modifying the proof of (5) which appears in [78]. ii) The first two upper bounds in (7) are tight. Moreover, the second inequality in (7) becomes an equalityif any n − of the K i ’s are tightly covered(iii) Recall that (cid:96) ∈ K denotes a line segment. If K is any convex body, then coin( K ⊕ (cid:96) ) = 4 N / ( K ) .(iv) Let the convex body K be the Minkowski sum of the convex bodies K , . . . , K n ∈ K d and Γ be as in part(i). Then coin( K ) ≤ inf λ ≤ (cid:81) ni =1 N λ ( K i )1 − λ ≤ (cid:81) ni =1 N Γ ( K i )1 − Γ < n (cid:89) i =1 coin( K i ) . (8)The notion of tightly covered convex bodies introduced in [25] plays a critical role in Theorem 4.9 (ii)-(iii). Definition 2.
We say that a convex body K ∈ K d is tightly covered if for any < λ < , K contains N λ ( K ) points no two of which belong to the same homothet of K with homothety ratio λ . In [25], it was noted that not all convex bodies are tightly covered (e.g., B is not), (cid:96) ∈ K is tightlycovered and so is the d -dimensional cube C d , for any d ≥
2. Do other examples exist?
Problem 2.
For some d ≥ , find a tightly covered convex body K ∈ K d other than C d or show that no suchconvex body exists. Since coin is a lower semicontinuous functional defined on the compact space K d , it is guaranteed toachieve its infimum over K d . It turns out that in addition to determining minimizers in all dimensions, wecan also find a maximizer in the planar case. Theorem 4.10. (i) Let d be any positive integer and K ∈ K d . Then coin( C d ) = 2 d +1 ≤ coin( K ) and thus d -cubes minimizethe covering index in all dimensions.(ii) If K is a planar convex body then coin( K ) ≤ coin( B ) = 14 . Since B maximizes the covering index in the plane, it can be asked if the same is true for B d in higherdimensions. Problem 3.
For any d -dimensional convex body K , prove or disprove that coin( K ) ≤ coin( B d ) holds. Since coin( B d ) = O (2 d d / ln d ) [25], a positive answer to Problem 3 would considerably improve the bestknown upper bound on the illumination number I ( K ) = O (4 d √ d ln d ) when K is a general d -dimensionalconvex body to within a factor √ d of the bound I ( K ) = O (2 d d ln d ) when K is o -symmetric. This gives usa way to closing in on the Illumination Conjecture for general convex bodies.If we replace the restriction γ m ( K ) ≤ / γ m ( K ) <
1, the resulting quantity is called the weak covering index , denoted by coin w ( K ). As the namesuggests, the weak covering index loses some of the most important properties of the covering index. Forinstance, no suitable analogue of Theorem 4.9 (iii) exists for coin w ( · ). As a result, we can only estimate theweak covering index of cylinders. Also the discussed aspects of continuity of the covering index seem to belost for the weak covering index. Last, but not the least, unlike the covering index we cannot say much atall about the maximizers and minimizers of the weak covering index.In the end, we would like to mention that fractional analogues of the covering index and the weak coveringindex were introduced in [26]. Just like fractional illumination number, we do not discuss these here due tolimitation of space. So far in Section 4.1-4.2, we have discussed some quantitative versions of the illumination number. Theaim of this section is to introduce a quantification of the X-ray number. This quantification has the addedadvantage of connecting the X-ray problem with the Tarski’s plank problem and its relatives (see [23, Chapter4]). 15iven a linear subspace E ⊆ E d we denote the orthogonal projection on E by P E and the orthogonalcomplement of E by E ⊥ . Given 0 < k < d , define a k -codimensional cylinder C as a set, which can bepresented in the form C = B + H , where H is a k -dimensional linear subspace of E d and B is a measurable setin E := H ⊥ . Given a convex body K and a k -codimensional cylinder C = B + H denote the cross-sectionalvolume of C with respect to K bycrv K ( C ) := vol d − k ( C ∩ E )vol d − k ( P E K ) = vol d − k ( P E C )vol d − k ( P E K ) = vol d − k ( B )vol d − k ( P E K ) . We note that if T : E d → E d is an invertible affine map, then crv K ( C ) = crv T ( K ) ( T ( C )). Now weintroduce the following. Definition 3 ( k -th Cylindrical Covering Parameter) . Let < k < d and K be a convex body in E d . Thenthe k -th cylindrical covering parameter of K is labelled by cyl k ( K ) and it is defined as follows: cyl k ( K ) = inf (cid:83) ni =1 C i (cid:26) n (cid:88) i =1 crv K ( C i ) : K ⊆ n (cid:91) i =1 C i , C i is a k − codimensional cylinder , i = 1 , . . . , n (cid:27) . We observe that if T : E d → E d is an invertible affine map, then cyl k ( K ) = cyl k ( T ( K )). Furthermore, itis clear that cyl k ( K ) ≤ K in E d and for any 0 < k < d . In terms of X-raying,one can think of cyl k ( K ) as the minimum of the ‘sum of sizes’ of ( d − k )-dimensional X-raying windowsneeded to X-ray K .Recall that a ( d − E d is also called a plank for the reason that it is the set ofpoints lying between two parallel hyperplanes in E d . The width of a plank is simply the distance between thetwo parallel hyperplanes. In a remarkable paper [6], Bang has given an elegant proof of the Plank Conjectureof Tarski showing that if a convex body is covered by finitely many planks in E d , then the sum of the widthsof the planks is at least as large as the minimal width of the body, which is the smallest distance betweentwo parallel supporting hyperplanes of the given convex body. A celebrated extension of Bang’s theorem to d -dimensional normed spaces has been given by Ball in [5]. In his paper [6], Bang raises his so-called AffinePlank Conjecture, which in terms of our notation can be phrased as follows. Conjecture 4.11 ( Affine Plank Conjecture). If K is a convex body in E d , then cyl d − ( K ) = 1 . Now, Ball’s celebrated plank theorem ([5]) can be stated as follows.
Theorem 4.12. If K is an o -symmetric convex body in E d , then cyl d − ( K ) = 1 . Bang [6] also raised the important related question of whether the sum of the base areas of finitely many(1-codimensional) cylinders covering a 3-dimensional convex body is at least half of the minimum area of a2-dimensional projection of the body. This, in terms of our terminology, reads as follows.
Conjecture 4.13 (1 -Codimensional Cylinder Covering Conjecture). If K is a convex body in E ,then cyl ( K ) ≥ . If true, then Bang’s estimate is sharp due to a covering of a regular tetrahedron by two cylinders describedin [6]. In connection with Conjecture 4.13 the first named author and Litvak have proved the following generalestimates in [21].
Theorem 4.14.
Let < k < d and K be a convex body in E d . Then cyl k ( K ) ≥ ( dk ) . Furthermore, it is proved in [21] that if K is an ellipsoid in E d , then cyl ( K ) = 1. Akopyan, Karasev andPetrov ([2]) have recently proved that if K is an ellipsoid in E d , then cyl ( K ) = 1. They have put forward: Conjecture 4.15 ( Ellipsoid Conjecture). If K is an ellipsoid in E d , then cyl k ( K ) = 1 for all < k < d . A computer-based approach
Given a positive integer m , Lassak [60] introduced the m -covering number of a convex body K as the minimalpositive homothety ratio needed to cover K by m positive homothets. That is, γ m ( K ) = inf (cid:40) λ > K ⊆ m (cid:91) i =1 ( λ K + t i ) , t i ∈ E d , i = 1 , . . . , m (cid:41) . Lassak showed that the m -covering number is well-defined and studied the special case m = 4 for planarconvex bodies. It should be noted that special values of this quantity had been considered by several authorsin the past. For instance, in the 70’s and 80’s the first named author showed that γ ( B ) = 0 . . . . and γ ( B ) = 0 . . . . [7, 8].Figure 6: Optimal configurations that demonstrate γ ( B ) = 0 . . . . and γ ( B ) = 0 . . . . Zong [88] studied γ m : K d → R as a functional and proved it to be uniformly continuous for all m and d . He did not use the term m -covering number for γ m ( K ) and simply referred to it as the smallest positivehomothety ratio. In [25], we proved the following stronger result. Theorem 5.1.
For any
K, L ∈ K d , γ m ( K ) ≤ d BM ( K, L ) γ m ( L ) holds and so γ m is Lipschitz continuous on K d with d −
12 ln d as a Lipschitz constant and | γ m ( K ) − γ m ( L ) | ≤ d BM ( K, L ) − ≤ d −
12 ln d ln ( d BM ( K, L )) , for all K, L ∈ K d . Further properties and some variants of γ m ( · ) are discussed in the recent papers [55, 83]. For instance,it has been shown in [83] that for any d -dimensional convex polytope P with m vertices, we have γ m ( K ) ≤ d − d . Obviously, any K ∈ K d can be covered by 2 d smaller positive homothets if and only if γ d ( K ) <
1. Zongused these ideas to propose a possible computer-based approach to attack the Covering Conjecture [88].Recall that in a metric space, such as K d , an (cid:15) -net ξ is a finite or infinite subset of K d such that theunion of closed balls of radius (cid:15) centered at elements of ξ covers the whole space. Thus if an (cid:15) -net exists,any element of K d is within Banach–Mazur distance (cid:15) of some element of the cover. The key idea of theprocedure proposed by Zong is the construction of a finite (cid:15) -net of K d whose elements are convex polytopes,for every real number (cid:15) > d . Here we describe the construction briefly.We first take an affine image of a d -dimensional convex body K that is sandwiched between the unit ball B d centered at the origin and the ball d B d with radius d . Such an image always exists by John’s ellipsoid Cover the Spot is a popular carnival game in the United States. The objective is to cover a given circular spot by 5 circulardisks of smaller radius. It seems that by determining γ ( B ), the first named author was unwittingly providing the optimalsolution for Cover the Spot ! Recall that the Banach–Mazur distance is a multiplicative metric and so the condition (cid:15) > (cid:15) > { C , . . . , C m } of the boundary of d B d with spherical caps C i as shownin Figure 5. The centers of the caps C i are joined to the origin by lines { L , . . . , L m } and a large number ofequidistant points are taken on the lines L i . We denote by p i the point lying in K ∩ L i that is farthest fromthe origin. Then the convex hull P = conv { p i } is the required element of our (cid:15) -net. Zong [88] showed thatby taking m large enough and increasing the number of points on L i we can ensure d BM ( K , P ) ≤ (cid:15) .Figure 7: Construction of an (cid:15) -net of K d He then notes that if we manage to construct a finite (cid:15) -net ξ = { P i : i = 1 , . . . , j } of K d , satisfying γ d ( P i ) ≤ c d for some c d < (cid:15) , then γ d ( K ) < K ∈ K d . Thiswould imply that the Covering Conjecture is true in dimension d .The following is a four step approach suggested by Zong [88]. Zong’s Program:
1. For a given dimension such as d = 3, investigate (with the assistance of a computer) γ d ( K ) for someparticular convex bodies K and choose a candidate constant c d .2. Choose a suitable (cid:15) .3. Construct an (cid:15) -net ξ of sufficiently small cardinality.4. Check (with the assistance of a computer) that the minimal γ d -value over all elements of ξ is boundedabove by c d .Indeed this approach appears to be promising and, to the authors’ knowledge, is a first attempt at acomputer-based resolution of the Covering Conjecture. However, Zong’s program is not without its pitfalls.For one, it would take an extensive computational experiment to come up with a good candidate constant c d . Secondly, Zong’s (cid:15) -net construction leads to a net with exponentially large number of elements. In fact,using B¨or¨oczky and Wintsche’s estimate [41] on the number of caps in a spherical cap covering, Zong [88]showed that | ξ | ≤ (cid:22) d ln (cid:15) (cid:23) α d d d +3 (ln (cid:15) ) − d , (9)where c is an absolute constant. Since Zong’s construction does not provide much room for improving theabove estimate, better constructions are needed to reduce the size of ξ , while at the same time keeping (cid:15) sufficiently small. Problem 4.
Develop a computationally efficient procedure for constructing (cid:15) -nets of K d of small cardinality. Addressing the above problem would be a critical first step in implementing Zong’s program. Wu [83](also see [55]) has recently proposed two variants of γ m ( · ) that can be used in Zong’s program instead.However, the challenges and implementation issues remain the same.18 cknowledgments The first author is partially supported by a Natural Sciences and Engineering Research Council of CanadaDiscovery Grant. The second author is supported by a Vanier Canada Graduate Scholarship (NSERC) andAlberta Innovates Technology Futures (AITF).
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K´aroly Bezdek
Department of Mathematics and Statistics, University of Calgary, CanadaDepartment of Mathematics, University of Pannonia, Veszpr´em, Hungary
E-mail:[email protected] andMuhammad A. Khan
Department of Mathematics and Statistics, University of Calgary, Canada