Fractional Planks

Discrete & Computational Geometry, Jan 2002

In 1950 Bang proposed a conjecture which became known as “the plank conjecture”: Suppose that a convex set S contained in the unit cube of R n and touching all its sides is covered by planks. (A plank is a set of the form {(x 1 , ..., x n ): x j ∈ I} for some j ∈ {1, ...,n} and a measurable subset I of [0, 1]. Its width is defined as |I| .) Then the sum of the widths of the planks is at least 1 . We consider a version of the conjecture in which the planks are fractional. Namely, we look at n -tuples f 1 , ..., f n of nonnegative-valued measurable functions on [0,1] which cover the set S in the sense that ∑ f j (x j ) ≥ 1 for all (x 1 , ..., x n )∈ S . The width of a function f j is defined as ∈t 0 1 f j (x) dx . In particular, we are interested in conditions on a convex subset of the unit cube in R n which ensure that it cannot be covered by fractional planks (functions) whose sum of widths (integrals) is less than 1 . We prove that this (and, a fortiori, the plank conjecture) is true for sets which touch all edges incident with two antipodal points in the cube. For general convex bodies inscribed in the unit cube in R n we prove that the sum of widths must be at least 1/n (the true bound is conjectured to be 2/n).

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Fractional Planks

Discrete Comput Geom Fractional Planks¤ Ron Aharoni 1 Ron Holzman 1 Michael Krivelevich 0 Roy Meshulam 1 0 Department of Mathematics, Sackler Faculty of Exact Sciences, Tel Aviv University , 69978 Tel Aviv , Israel 1 Department of Mathematics , Technion, 32000 Haifa , Israel In 1950 Bang proposed a conjecture which became known as “the plank conjecture”: Suppose that a convex set S contained in the unit cube of <n and touching all its sides is covered by planks. (A plank is a set of the form f.x1; : : : ; xn /: xj 2 I g for some j 2 f1; : : : ; ng and a measurable subset I of [0; 1]. Its width is defined as jI j.) Then the sum of the widths of the planks is at least 1. We consider a version of the conjecture in which the planks are fractional. Namely, we look at n-tuples f1; : : : ; fn of nonnegative-valued measurable functions on [0; 1] which cover the set S in the sense that P fj .xj / ¸ 1 for all .x1; : : : ; xn / 2 S. The width of a function fj is defined as R 1 0 fj .x / d x . In particular, we are interested in conditions on a convex subset of the unit cube in <n which ensure that it cannot be covered by fractional planks (functions) whose sum of widths (integrals) is less than 1. We prove that this (and, a fortiori, the plank conjecture) is true for sets which touch all edges incident with two antipodal points in the cube. For general convex bodies inscribed in the unit cube in <n we prove that the sum of widths must be at least 1=n (the true bound is conjectured to be 2=n). ¤ Ron Aharoni's research was supported in part in MIPG at UPENN by NIH Grant Number HL28438, and by the fund for the promotion of research at the Technion. Ron Holzman's research was supported by the fund for the promotion of research at the Technion. - In 1950 Bang [4], [5] proved the following conjecture of Tarski: if a convex body of width 1 in <n is covered by slabs, then the sum of the widths of the slabs is at least 1 (in other words, the most economical way to cover the body is by just one slab). He then asked the following, more demanding question. Let S be a convex body in <n, and let T1; : : : ; Tm be slabs whose union covers S. Is it true that the sum of the relative widths of the slabs is at least 1? (The relative width of the slab with respect to S is the ratio of the width of the slab to the width of S in the same direction.) This conjecture gained the name “the plank conjecture,” for obvious reasons. It has a number of equivalent formulations, including a geometric pigeonhole principle suggested by Davenport and generalized by Alexander [2]. For the version of the conjecture which will be used here we need a few definitions. The unit cube in <n will be denoted by Qn (to avoid trivial exceptions, we assume throughout that n ¸ 2). A subset of Qn is called packed in the cube if it touches all facets (sides) of the cube. A plank P (of type j ) in Qn is a set of the form fx 2 Qn: xj 2 I g for some measurable subset I of [0; 1]. (The notation x will henceforth be reserved for a point .x1; x2; : : : ; xn/ in Qn.) We then write P D Plj .I /. The width j Pj of the plank is jI j, the Lebesgue measure of I . Given a subset S of Qn, a family P D . P1; : : : ; Pn/ of planks (where Pj is of type j ) is called a plank cover of S if its union contains S. The total width jPj of the family is the sum of the widths of the planks in it. Conjecture 1.1 (The Plank Conjecture). A plank cover of a packed convex set in Qn has total width at least 1. This conjecture is known to be true for n D 2 [6] and when the set is centrally symmetric [3]. We note that it is customary to reserve the term “plank” for the case when the set I (which we allow to be an arbitrary measurable subset of [0; 1]) is an interval. It is straightforward to reduce our formulation of the plank conjecture to one which uses only interval planks, but allows any number of them in each direction. A plank of type j can be viewed as a 0; 1 function fj .xj / on the interval [0; 1]. Viewing planks this way, it is natural to ask what happens when each plank Plj is replaced by a nonnegative real-valued measurable function fj .xj /, instead of a 0; 1 function. The covering condition is then that P fj .xj / ¸ 1 for each point x 2 S. If this condition holds then we say that the system f1; : : : ; fn is a fractional plank cover of S. Notation. For a measurable function f from < to < and a measurable subset T of its domain, we denote by f .T / the integral RT f .x / d x . If T is an interval [a; b], we abbreviate and write f [a; b] for f .[a; b]/. The width j fj j of a fractional plank fj is fj [0; 1]. Given a system f1; : : : ; fn of fractional planks, the total width of the system is defined as P j fj j. The infimum of the total width over all fractional plank covers of S is called the fractional plank covering number of S. By standard arguments (see Theorem 2.21 of [9]) this infimum is attained, that is, it is a minimum. This minimum is denoted by ¿ ¤.S/. The source of this notation is in combinatorics. To explain it, we need the following terminology (a standard reference for which is, say, [7]). A hypergraph is a family H of subsets (called edges) of some ground set (whose elements are called vertices). A fractional cover of H is a system of nonnegative real weights on the vertices which sums up to at least 1 on each edge. The minimal sum of weights over all fractional covers is called the fractional covering number of H , and is denoted by ¿ ¤.H /. Now, a subset S of Qn can be viewed as a hypergraph H D H .S/. The ground set is the disjoint union of n copies of [0; 1], and each point x 2 S corresponds to an edge of H .S/, namely the subset fx1; x2; : : : ; xng of that union. The fractional plank covering number is thus a continuous version of the fractional covering number of the hypergraph. In this terminology the plank covering number of S (that is, the infimum of the total width over all plank covers of S) is the analogue of the “covering number” of the hypergraph, which is the minimal number of vertices which meet all edges. Hence it is appropriate to assign to it the usual notation for this parameter, namely ¿ .S/. (We believe, but cannot prove, that ¿ .S/ is in fact a minimum, that is, that there exists a plank cover attaining it, for every measurable subset S of Qn.) Since the notion of fractional plank covers is more general than that of ordinary plank covers, ¿ ¤ · ¿ , and in fact usually strict inequality obtains. Thus there are packed convex sets in Qn with ¿ ¤ < 1. In fact, ¿ ¤ may be as low as 2=n for such sets, as the following example shows: Denote by 1n the standard .n ¡ 1/-dimensional simplex in <n, namely the set of all points x 2 Qn such that P xj D 1. Let fj .x / D .2=n ¡ x /C . j D 1; : : : ; n/, that is: fj .x / D 2=n ¡ x for 0 · x · 2=n, fj .x / D 0 for 2=n < x · 1. Then, for each point x 2 1n we have The total width of the system is P fj [0; 1] D 2=n, implying that ¿ ¤.1n/ · 2=n. Later we shall see that, in fact, ¿ ¤.1n/ D 2=n, and that if the plank conjecture is true then ¿ ¤ ¸ 2=n for all packed convex sets in Qn. 2. A Dual Concept A measure matching on a measurable subset S of Qn is a nonnegative measure defined on the Lebesgue-measurable subsets of Qn, whose support is contained in S, and whose marginal measure on each of the coordinates has density at most 1. That is, the measure of any plank of width ± is at most ±. (A similar concept, confined to probability measures, was introduced by Gardner [8] as a tool for studying the plank conjecture; he used the term “relative width measure for the coordinate directions.”) In the discrete case this corresponds to a system of nonnegative real weights on the points of S, such that for each coordinate j and each u 2 [0; 1] the sum of the weights of the points x satisfying xj D u is at most 1. The name used in combinatorics for such a system is a fractional matching of the hypergraph represented by S. The supremum of ¹.S/ over all measure matchings ¹ on S is called the measure matching number of S, and is denoted (following the combinatorial convention) by º¤.S/. By standard measure theoretical arguments (see Theorem 2.19 of [9]) it follows that if S is compact, then º¤.S/ is attained. Given a measure matching ¹ on S with marginals ¹1; : : : ; ¹n and a fractional plank cover f1; : : : ; fn, one clearly has ¹.S/ · Z X fj .xj / d¹.x/ D S j X Z 1 j This implies that º¤.S/ · ¿ ¤.S/. An analogue of the duality theorem of linear programming [9, Corollary 2.18] yields that º¤ D ¿ ¤ for all measurable subsets of Qn. 3. A Fractional Version of the Plank Conjecture A hypergraph is called n-partite if it admits a partition of the vertex set into n parts, such that every edge consists of a choice of one vertex from each part. In [10] Lova´sz proved that for n-partite hypergraphs the inequality ¿ · .n=2/¿ ¤ holds. Since, as noted above, subsets of Qn may be viewed as n-partite hypergraphs, the same inequality holds for them, too. Theorem 3.1. for any measurable subset S of Qn. n ¿ .S/ · 2 ¿ ¤.S/ The proof below is a continuous version of Lova´sz’ proof. At its base lies the following observation, which is a special case of Proposition 2 in [1]: Lemma 3.2. There exists a measure matching ¹ on 1n whose support is contained in the intersection of 1n with the cube fx: 0 · xj · 2=n; 1 · j · ng, and whose marginal measure in each direction xj has density 1 on the interval 0 · xj · 2=n. Proof. The lemma is trivial for n D 2. We first prove the case n D 3. There are many ways of constructing a measure on 13 which do the job. One of them uses the following: Theorem 3.3 (Archimedes). On the unit disk there exists a positive measure with constant marginals in all directions. In fact, the measure is given by a function: the function 1 p1 ¡ kx k2 has the property that its integrals on intersections of lines with the unit disk are all equal. Applying the theorem to the disk inscribed in the triangle 13, and normalizing the measure suitably, yields the desired measure. As the lemma is true for n D 2; 3, to prove it in general it suffices to note that if it is true for two values ` and m of n, then it holds also for ` C m. Indeed, the Cartesian product ` m ` C m 1` £ ` C m 1m naturally embeds in 1`Cm . On each of the factors of this product we have, by assumption, a suitable measure, and the product of those measures satisfies the requirements on 2 Remark. The lemma shows that º¤.1n/ ¸ 2=n, and combining this with the reverse inequality proved in the Introduction, º¤.1n/ D ¿ ¤.1n/ D 2=n for n ¸ 2. Proof of Theorem 3.1. Let f1; f2; : : : ; fn be a fractional plank cover for the set S, with total width w. We prove that there exists a plank cover with total width at most .n=2/w. For each point x 2 1n we define a plank cover for S, as follows. For each 1 · j · n let Ij D Ij .x/ be the set of points x 2 [0; 1] for which fj .x / ¸ xj . Let Pj D Plj .Ij /. Since P xj D 1, and since P fj .sj / ¸ 1 for all points s D .s1; s2; : : : ; sn/ 2 S, it follows that for each point s 2 S there exists a j · n for which s 2 Pj , i.e., P.x/ D P1; : : : ; Pn is a plank cover. In order to show that there exists a plank cover of width at most .n=2/w it suffices to show that the normalized ¹-average over all x 2 1n of the widths of P.x/ is at most that number (where ¹ is a measure on 1n satisfying the requirements of Lemma 3.2). For each j we have Z j Pj j d¹ D jfx : fj .x / ¸ xj gj d¹ Z Z 2=n 0 D jfx : fj .x / ¸ ygj d y D Z 1 0 min ½ 2 n ¾ ; fj .x / d x · j fj j: (The second equality follows from the equidistribution of the marginal of ¹ in the j th coordinate, between 0 and 2=n. The third equality is a general property of integrals.) Thus R jP.x/j d¹ · P j fj j D w, and since ¹ is of total weight 2=n, it follows that the normalized ¹-average of jPj is at most .n=2/w, as promised. 2 By Theorem 3.1, if the plank conjecture is true, then the following conjecture also holds: Conjecture 3.4. ¿ ¤.S/ ¸ 2=n for any convex set S which is packed in Qn . As already noted, the simplex 1n is an example where equality holds in Conjecture 3.4. In fact, the simplex is but one member of a family of sets in which equality obtains. We name the members of this family “generalized octahedra,” and they are defined as follows. Let c D .c1; c2; : : : ; cn/ be a point in Qn, and let P.c/ be the set of projections of c on the 2n facets of Qn. Any convex set S which satisfies conv. P.c/nfcg/ µ S µ conv. P.c// is called a generalized octahedron, with center c. (Here and elsewhere conv. A/ denotes the convex hull of the set A. The definition is devised so as to cope with the special case in which the center is a vertex of the cube, in which case it is one of its own projections. This includes the case of the simplex, but also that of the body consisting of the simplex together with all points below it, i.e., the convex hull of the simplex and the vertex of the cube which it encloses.) Generalized octahedra play a special role with respect to ¿ and ¿ ¤. First, as already mentioned, they satisfy ¿ ¤ D 2=n, namely they are extreme cases in Conjecture 3.4. We suspect that these are the only extreme cases for n ¸ 3. Generalized octahedra are also the only packed convex sets we know, in which there exists a family of more than one plank covering the set, with total width 1 (i.e., these are the only cases known in which the bound 1 in the plank conjecture is attained in a nontrivial way). Still another fact about generalized octahedra is that in the case n D 2 they are the only packed convex sets in Q2 in which measure matchings, rather than functions, are really needed. That is, we can prove that these are the only packed convex sets in the square for which no measure matching with mass 1 on the set can be represented as the integral of a function with respect to the two-dimensional Lebesgue measure. In the original plank conjecture no dimension-independent constant lower bound is known. In contrast, Conjecture 3.4 can be proved to within a factor of 2. In fact, we prove a little more: not only that ¿ ¤ of every convex set packed in Qn is at least 1=n, but also º, the matching number of the set, is at least 1=n. However, first we have to define this notion: Definition 3.5. Given a segment T in <n, we denote by º.T / the minimum among the lengths of its projections on the axes. Definition 3.6. The matching number º.S/ of a subset S of Qn is the supremum of PT 2T º.T /, where T ranges over all finite families of segments contained in S, whose projections on all axes are pairwise disjoint. This definition is obtained from the standard definition of the matching number of hypergraphs, by discretization. Note that the supremum in this definition is not always attained, not even if we allow infinite families of segments. An example in which the supremum is not attained is the set of points in Q2 satisfying the inequality y ¸ 2jx ¡ 12 j. The matching number of this set is 1, but it has no matching in the sense of Definition 3.6 with full projections. It is easily seen that º.S/ · º¤.S/: put a uniform measure with total mass º.T / on each segment T 2 T . Hence the following theorem implies that ¿ ¤.S/ ¸ 1=n for all packed convex subsets S of Qn: Theorem 3.7. º.S/ ¸ 1=n for every convex set S which is packed in Qn. Proof. Without loss of generality we may assume that S is closed. The difference body K D S ¡ S of S is then convex, compact, centrally symmetric, and packed in the cube [¡1; 1]n. As noted above, Ball [3] proved the plank conjecture for such bodies. Hence, denoting by Dj the open plank Plj ..¡1=n; 1=n// .1 · j · n/, the set K is not contained in the union of the planks Dj (whose sum of widths is 2). (We have also used here the compactness of K .) Thus there exists a point k D u ¡ v 2 K not belonging to any Dj , where u; v 2 S. The segment whose endpoints are u and v is contained in S, and its projections on all axes are all no shorter than 1=n. Remark 1. The convex hull of the midpoints of all facets of the cube shows that 1=n is the best lower bound possible on º of a single segment contained in a packed convex body in Qn. Remark 2. A conjecture of Ryser and Lova´sz (see p. 105 of [7]) states that ¿ · .n¡1/º for any n-partite hypergraph. Combining this with the plank conjecture would yield that º.S/ ¸ 1=.n ¡ 1/ for every convex set S which is packed in Qn. However, it is quite possible that this is not the best bound for n ¸ 3. Indeed, 1n, which we conjecture to be extreme for ¿ ¤, has º > 1=.n ¡ 1/. We calculate it for n D 3: 3 Proposition 3.8. º.13/ D 5 . Proof. We prove here only one direction, namely we construct a matching in 13 with º D 35 . The other direction is rather complicated, and is omitted. Let p1 D .0; 15 ; 45 /; q1 D . 15 ; 25 ; 25 /, and let p2; p3 be the two cyclic permutations of p1, and q2; q3 the two cyclic permutations of q1. Finally, let Ij . j D 1; 2; 3/ be the segments joining pj with qj . Then it is easy to check that the projections of the three segments on each axis have disjoint interiors, while º.Ij / D 51 , yielding º.13/ ¸ 5 . 3 4. Hefty Sets One aim of this paper is to study conditions which imply that a subset of Qn has ¿ ¤ D 1. Definition 4.1. A measurable subset S of Qn is called hefty if ¿ ¤.S/ D 1. Proposition 4.2. The following four conditions on a measurable subset S of Qn are equivalent: (1) S is hefty. (2) There exists a probability measure on S with uniform marginals (i.e., the marginal on each axis is the ordinary Lebesgue measure on [0; 1]). (3) Given any system ¯1; ¯2; : : : ; ¯n of measurable functions on [0; 1], if P ¯j .xj / ¸ 1 for each point .x1; x2; : : : ; xn/ 2 S, then P ¯j [0; 1] ¸ 1. (4) Given any system ®1; ®2; : : : ; ®n of measurable functions on [0; 1], if P ®j .xj / ¸ 0 for each point .x1; x2; : : : ; xn/ 2 S, then P ®j [0; 1] ¸ 0. Proof. The equivalence between (1) and (2) follows from Section 2. That (3) implies (1) is clear, since the conclusion in both is the same, while the condition in (1) is stronger— there is the additional condition that the functions are nonnegative valued. The equivalence of (3) and (4) is also easy: assuming that (3) holds, given a system of functions ®j as in (4) define ¯1 D ®1 C 1; ¯j D ®j for j > 1. Applying (3) to the system ¯j then yields (4). The reverse implication is similar. It remains to show that (2) implies (3). Assume that there exists a probability measure ¹ on S with uniform marginals. Let ¯1; : : : ; ¯n be functions as in (3). Write the analogue of (1) for the functions ¯j : 1 D ¹.S/ · Z X ¯j .xj / d¹.x/ D S j X Z 1 j 0 ¯j .xj / d¹j .xj / D Note that while in (1) the nonnegativity of the functions fj was needed to obtain the right-hand inequality, here it is not necessary that the functions ¯j be nonnegative, since the marginals ¹j are known to be equal to the Lebesgue measure. 5. The Special Role of the Center Notation. For a real number t we denote by Et the vector whose entries are all t (the dimension of the vector is to be understood from the context). It turns out that the center of the cube, the point E12, plays a special role with regard to heftiness: Proposition 5.1. A closed convex hefty set contains E12 . Proof. If S is closed and convex and E12 62 S, then there exist real numbers r1; : : : ; rn and a positive " such that P rj xj ¸ P 12 rj C " for all points x 2 S. Applying condition 1 (4) of Proposition 4.2 to the functions ®j .xj / D rj .xj ¡ 2 / ¡ "=n yields a contradiction to the heftiness of S. The converse of the proposition is false. Even the ball Bn inscribed in Qn is not hefty, for n ¸ 4. (B3 is hefty, a fact which was already known to Archimedes—the measure matching showing this is similar to that appearing in the proof of Theorem 3.3.) The following proposition settles in the negative a problem of Gardner [8]: Proposition 5.2. For n ¸ 4 the ball Bn is not hefty. Proof. A point x 2 Bn satisfies Pi .xi ¡ 21 /2 · 41 , which implies, by the Cauchy– Schwartz inequality, Hence, defining for every 1 · i · n, we have X i 1 jxi ¡ 2 j · pn 2 : fi .x / D r 1 1 ; 4n ¡ jx ¡ 2 j X fi .xi / ¸ However, as is easily seen, P fi [0; 1] D pn=2 ¡ n=4, the last expression being negative for n > 4 and 0 for n D 4. Thus, for n > 4 the proposition follows from Proposition 4.2. In the case n D 4, note that equality is attained in (2) only at a finite number of points. Hence by changing one of the functions fi .x / slightly in a small neighborhood of some value of xi such that there is no point with equality having that value of xi , we can still maintain property (2), while having a negative sum of integrals. The following proposition presents a case in which the converse of Proposition 5.1 is true: Proposition 5.3. If a set of vertices of Qn has E12 in its convex hull, then the convex hull is hefty. Proof. Let V D fvk : k 2 K g be the set of vertices in question, with vk D .v1k ; : : : ; vnk /. Write E12 D Pk2K ®k vk , where ®k ¸ 0 and P ®k D 1. For each k 2 K let Tk be the segment connecting vk with E12, and distribute uniformly on Tk a measure with total mass ®k . Let ¹ be the measure on conv.V / which is concentrated on the union of the segments Tk and is the sum of all the above measures. We shall show that ¹ is a measure matching: this will complete the proof, since obviously the mass of ¹ is 1. Let 1 · j · n. Since Pf®k : vjk D 1g D Pf®k : vjk D 0g D 12 , the mass of ¹ on Plj [0; 12 ] is 12 . Since ¹ is evenly distributed on each segment Tk , it follows that this mass of 12 is evenly distributed on [0; 12 ], which means that ¹. Plj .I // D jI j for every interval I ½ [0; 12 ]. The same is true for all subintervals of [ 12 ; 1], which implies the desired conclusion. Another result in the converse direction to that of Proposition 5.1 was proved in [1]. A subset of Qn is called hexagonal if it is packed, and is the intersection of a hyperplane with Qn. (The source of the name is that in the case n D 3 such a set has a (possibly degenerate) hexagonal shape.) Theorem 5.4. A hexagonal set containing E12 is hefty. Since, in fact, Proposition 2 in [1] is formulated a bit differently, we give here an outline of the proof. The proof uses the following easy observation (which is needed if one wants to translate Proposition 2 of [1] into the terms of Theorem 5.4). Lemma 5.5. (i) If a hyperplane a1x1 C a2x2 C ¢ ¢ ¢ C an xn D c meets all facets of Qn , then jai j · X j6Di jaj j .3/ for all 1 · i · n. (ii) If a hyperplane a1x1 C a2x2 C ¢ ¢ ¢ C an xn D c passes through E12 and satisfies (3), then it meets all facets of the cube. Outline of Proof of Theorem 5.4. For n D 3 it is possible to provide concretely a measure matching with total mass 1 on the hexagon, concentrated on certain of its diagonals. The case n > 3 is done by induction on n. Let X be a hexagonal set, namely X is the set of points x 2 Qn satisfying the equation a1x1 C a2x2 C ¢ ¢ ¢ C an xn D c. Without loss of generality, assume that 0 · a1 · a2 · ¢ ¢ ¢ · an. Now “join” the first two variables, i.e., look at the hyperplane H in <n¡1 defined by .a1 C a2/y C Pi>2 ai xi D c. Since n > 3, H satisfies (3). Hence, by the induction hypothesis, the intersection of H with Qn¡1 is hefty, and the probability measure on this intersection readily yields a probability measure on the original hexagonal set. 2 6. Other Conditions Implying Heftiness As already mentioned, the plank conjecture is true for n D 2. Since the combinatorial interpretation of this case is that of bipartite graphs, and since for such graphs º D ¿ ¤ D ¿ , this implies that all convex sets packed in Q2 are hefty. (Gardner [8] reached the same conclusion by constructing measure matchings for such sets.) As noted above, for n > 2 this is no longer true. It is tempting to ascribe the difference between the two cases to the fact that in the case n D 2 “facets” and “edges” coincide, and thus being packed means not only touching the facets of the cube, but also the edges. Indeed, it is not hard to prove the following: Proposition 6.1. A convex subset of Qn touching all of its edges is hefty. In fact, a much weaker condition (though equivalent in the case n D 2) suffices: Definition 6.2. A subset S of Qn is called strongly packed if there exist two antipodal vertices of Qn such that S touches all 2n edges incident with them. The main theorem of this paper is: Theorem 6.3. A strongly packed convex subset of Qn is hefty. In order to understand the intuition behind this theorem, note, first, that a subset of Qn containing a main diagonal (a segment connecting two antipodal vertices) is hefty (this will be shown below). The idea behind the theorem is that the property of having a hefty convex hull is preserved upon replacing each of these two vertices by n points on the edges incident with it. That is, the n “splinters” of the vertex do the work that the single vertex had done. In fact, we believe that this is true in general, namely: Conjecture 6.4. If a vertex v of the cube belongs to a set S having the property that its convex hull is hefty, then replacing v by n points on the edges incident with it preserves this property of S. It will also be useful to have a name for the operation in this conjecture: Definition 6.5. A replacement as in the conjecture is called a splitting of v. For the property of having E12 in the convex hull, the analogue of Conjecture 6.4 is indeed true: Proposition 6.6. If E12 2 conv.S/ and S0 is obtained by splitting a vertex in S, then 1 E2 2 conv.S0/. Proof. We may assume that the split vertex is 0E. Then there exists a point u 2 conv.Snf0Eg/ such that E12 is on the segment connecting 0E and u. Let z be the point on the line connecting 0E and E12 which lies on the hyperplane spanned by the n splinters of 0E. Clearly, the order of the points on the line .t; t; : : : ; t / is 0E; z; E1; u, and thus E12 is on 2 the segment connecting z and u, and thus is in the convex hull of S0. The remainder of the paper is mainly devoted to the proof of Theorem 6.3. The main tool used in the proof is a certain property of families of sub-boxes of Qn, which we study in the next section. 7. ¿ ¤-Determining and Volume-Determining Families of Boxes A main tool in our investigations is that of ¿ ¤-determining families of boxes. These are families of sub-boxes of Qn having the property that if a given set S is hefty relative to every box in the family, then S is hefty in Qn. We find a surprising equivalent condition: that the boxes cannot all be enlarged simultaneously by infinitesimal changes in their boundaries. Here are precise definitions of these notions. By a box in <n we always mean a box whose sides are parallel to the axes. For such a box B we denote by Ij .B/ .1 · j · n/ the interval which is the projection of B on the xj -axis, and by `j .B/ the length of Ij .B/. Unless otherwise stated, we assume that boxes are not degenerate, namely `j .B/ > 0 for all j . By vol.B/ we denote the volume of B. Given a box B and a system F D . f1; f2; : : : ; fn/ of measurable real-valued functions, where the domain of fj contains Ij .B/, we write t .B; F / D Pj fj .Ij .B//=`j .B/. This is a “normalized” version of the linear functional P fj [0; 1] which is used in the definition of ¿ ¤ (and, indeed, coincides with it when B D Qn). If the functions in F are nonnegative valued, and if Pj fj .xj / ¸ 1 for each point x 2 S \ B, then we say that F is a fractional plank cover of S relative to B. The minimum of t .B; F / over all fractional plank covers of S relative to B will be denoted by ¿B¤ .S/. An equivalent definition is that ¿B¤ .S/ is ¿ ¤ of the subset of Qn obtained by stretching S \ B by a factor of 1=`j .B/ in the direction of each coordinate j . Definition 7.1. A subset S of Qn is called B-hefty if ¿B¤ .S/ D 1. Definition 7.2. A family B of sub-boxes of Qn is ¿ ¤-determining if for each family F D . f1; : : : ; fn/ of nonnegative-valued measurable functions the inequalities t .B; F / ¸ 1; B 2 B, imply the inequality t .Qn; F / ¸ 1. Remark. Similarly to Proposition 4.2, it is easy to show that equivalently one can remove the nonnegativity condition, while requiring that t .B; F / ¸ 0; B 2 B, imply t .Qn; F / ¸ 0. The definition implies: Lemma 7.3. If a family B of sub-boxes of Qn is ¿ ¤-determining, then any set which is B-hefty for all boxes B 2 B is hefty. Let B be a finite family of boxes in <n. We write M.B/ for the smallest box containing all boxes in B. We denote by P.B/ the set of hyperplanes perpendicular to the axes and supporting the boxes in B. For each j , these hyperplanes partition Ij .M.B// into m j .B/ intervals Ijk ; 1 · k · m j .B/, each being of length (say) `kj . Let K j .B/ denote the set of those k for which the box B contains points with xj 2 Ijk (i.e., Ij .B/ is the union of the intervals Ijk ; k 2 K j .B/). Assigning a variable wkj to each interval Ijk , we can define a linear form tB.B/.w/ by tB.B/.w/ D X j PPkk22KKjj..BB//w`kjkj : Using this terminology, we obtain the following characterization of a ¿ ¤-determining family. Lemma 7.4. Let B be a family of sub-boxes of Qn. Let BO D B [ fQng. Then the following are equivalent: (i) B is ¿ ¤-determining. (ii) t O .Qn/ is a convex combination of t O .B/; B 2 B. B B (iii) t O .Qn/ is a nonnegative linear combination of t O .B/; B 2 B. B B Two families of boxes C and D will be called similar if m j .C/ D m j .D/ for each 1 · j · n, and if there is a bijection ': C ! D such that K j .'.C // D K j .C /; j D 1; : : : ; n, for all C 2 C. An equivalence class of the relation of similarity will be called a configuration of boxes. A configuration of boxes 0 will be called volume-determining if for every two families of boxes C; D 2 0, if vol.C / · vol.'.C // for all C 2 C (where ' is as above), then vol.M.C// · vol.M.D//. A configuration of boxes 0 will be called ¿ ¤-determining if every C 2 0 satisfying M.C/ D Qn is ¿ ¤-determining. A somewhat surprising fact is that these two conditions are equivalent: Theorem 7.5. A configuration of boxes is ¿ ¤-determining if and only if it is volumedetermining. Proof. Let 0 be a volume-determining configuration of boxes in <n, choose B 2 0, and let C D M .B/. Consider the expressions vol.B/; B 2 B, as functions of the variables `kj (the lengths of the intervals in the partitions induced by P.B/). By the assumption that 0 is volume-determining, for each vector v in <6j mj .B/, if all functions vol.B/, B 2 B, increase in the direction of v, then so does the function vol.C / D Qj Pk `kj . It follows that for all v, if rvol.B/ ¢ v > 0 for each B 2 B, then rvol.C / ¢ v ¸ 0. Replacing v by v C "E for " > 0 and letting " tend to 0, it follows that already rvol.B/ ¢ v ¸ 0 for each B 2 B implies rvol.C / ¢ v ¸ 0. By the theory of linear inequalities this means that rvol.C / is a nonnegative combination of the vectors rvol.B/; B 2 B. When C D Qn, this is easily seen to be equivalent to condition (iii) in Lemma 7.4. For the proof of the converse, assume that the configuration 0 is ¿ ¤-determining, and let C; D be two families in 0 with a correspondence ': C ! D between them. We have to show that if vol.C / · vol.'.C // .4/ for every C 2 C, then vol.M .C// · vol.M .D//. Without loss of generality, we assume that M .C/ D Qn. For each C 2 C let p.C / D à Pk2Kj .C/ `0 kj ! P k k2Kj .C/ `j 1· j·n ; where `kj D `kj .C/ and `0 kj D `kj .D/. Inequality (4) can be written as Yn Pk2Kj .C/ `0 kj jD1 Pk2Kj .C/ `kj ¸ 1: However, this means that the points p.C /; C 2 C, belong to the convex subset AH (the notation standing for “above hyperbola”) of the positive orthant of <n, consisting of those points whose product of coordinates is at least 1. Apply now condition (ii) of Lemma 7.4 to the family C, substituting `0 kj for wkj . The conclusion obtained is that the point .Pk `0 kj /1· j·n is a convex combination of the points p.C /, and since AH is convex, this point belongs to AH. However, this means precisely that vol.M .D// ¸ 1 D vol.M .C//, as required. The proof yields a little more: it suffices to consider volume changes corresponding to infinitesimal changes in the partitions. This leads to: Corollary 7.6. A family B of sub-boxes of Qn satisfying M .B/ D Qn is ¿ ¤-determining if and only if one cannot move the hyperplanes in P.B/, while fixing those supporting Qn itself, in such a way that all boxes in B strictly grow in volume. Corollary 7.7. If B is a family of sub-boxes of Qn satisfying M .B/ D Qn whose configuration is ¿ ¤-determining, then S B D Qn (we say then that B is covering). Proof. Assume that all boxes in B miss some cell C in the partition of Qn formed by the hyperplanes in P.B/. Then one can move the boundaries of C so as to make C fill almost all of Qn, while all boxes in B get smaller. This contradicts Corollary 7.6, applied to the new family of boxes. Note that a family of sub-boxes which is ¿ ¤-determining does not have to be covering: take the two boxes x ; y · 21 and x ; y ¸ 21 in Q2. Here is an example, in Q2, of a configuration which is covering but not ¿ ¤-determining: choose two points .a; b/ and .c; d/ in the unit square, where a < c and b < d. Let B consist of the four boxes 0 · x · c; 0 · y · dI a · x · 1; b · y · 1I 0 · x · a; d · y · 1; and c · x · 1; 0 · y · b. However, we conjecture that for all n, if m j .B/ D 2 for each j , then the condition is sufficient. Another simple corollary of Theorem 7.5, which can also be proved directly, is: Corollary 7.8. If the family of boxes B partitions Qn, then it is ¿ ¤-determining. We mainly use one particular ¿ ¤-determining configuration of boxes: Lemma 7.9. Let a1; a2; : : : ; an be numbers in the open interval .0; 1/, and let B0 D Qn jD1[0; aj ], Bk D Plk .[ak ; 1]/ .1 · k · n/: Then the family Bi ; 0 · i · n, is ¿ ¤determining. Proof. Enlarging the volumes of the boxes Bi ; 1 · i · n, means making the ai ’s smaller, which means making B0 smaller. By Corollary 7.6 the result follows. It is also not hard to find the coefficients of the expression of t O .Qn/ as a combination B of the tBO .Bi /’s: Note that t O .B0/.w/ D X B j w1j ; aj tBO .Bi /.w/ D 1 w¡i2ai C X.wk1 C wk2/ k6Di .1 · i · n/: Hence we have à ! Y aj t O .B0/.w/ C j B where ® is some positive number. n X.1 ¡ ai / iD1 à ! Y ak t O .Bi /.w/ B k6Di ! à ! D à X 1 j aj ¡ .n ¡ 1/ Y aj j X.w1j C w2j/ D ®t O .Qn/.w/; j B 8. Proof of Theorem 6.3 8.1. Splitting One Vertex We start with a basic observation, already used in the special case B D Qn: Proposition 8.1. A set containing a main diagonal of B is B-hefty. Proof. Let S be a set containing a main diagonal D of B, and let F be a fractional plank cover of S relative to B. Let xj .t / D aj C t .bj ¡ aj / .0 · t · 1/ range over Ij .B/ in such a way that .x1.t /; : : : ; xn.t // ranges over D. Since P fj .xj / ¸ 1 for every x 2 S \ B, we have P R 1 0 fj .xj .t // dt ¸ 1. However, this means that t .B; F / ¸ 1. We first prove the theorem in the case that just one of the two antipodal vertices is split. In this case the theorem is: Theorem 8.2. Let a1; : : : ; an be real numbers between 0 and 1, and let pj D aj ej , where ej is the j th unit vector, .0; : : : ; 1; : : : ; 0/. Then the set S D convfp1; : : : ; pn; 1Eg Proof. Let B0; B1; : : : ; Bn be the sub-boxes of Qn as defined in Lemma 7.9, and let F D . f1; : : : ; fn/ be a system of nonnegative-valued functions covering S. We have to prove that P fj [0; 1] ¸ 1. For each 0 · s · 1 let '.s/ D Pj fj .s/. For each pair i 6D j of coordinates let Di j be the degenerate box having the segment between the points pi and pj as a main diagonal. Note that each Bi ; 1 · i · n, has a main diagonal contained in S, and thus S is Bi hefty for 1 · i · n. What is missing for an application of Lemma 7.9 is the B0-heftiness of S. This, as it turns out, is true for n D 3 (the proof of this fact requires the theorem itself!) but fails in general for n > 3. However, the diagonals of Di j are contained in S, and t .B0; F / can be nicely expressed by t .Di j ; F / D Pk6Di; j fk .0/ C fi [0; ai ]=ai C fj [0; aj ]=aj . We have 1 X t .Di j ; F / ¡ t .B0; F / D n ¡ 1 i< j n ¡ 2 2 n '.0/ ¸ 2 ¡ n ¡ 2 '.0/: 2 This implies that if '.0/ · 1, then we are done. On the other hand, if '.s/ > 1 for all 0 · s · 1, then Pj fj [0; 1] D '[0; 1] > 1. This leads us to consider s0 D inffs: '.s/ · 1g. If the point sE0 D .s0; s0; : : : ; s0/ lies on or above the hyperplane H determined by p1; : : : ; pn, then the segment [sE0; 1E] is contained in S, and hence Pj fj [0; 1] D '[0; s0] C '[s0; 1] ¸ s0 C 1 ¡ s0 D 1. Thus we may assume that sE0 lies below H . Let r ¸ s0 be such that '.r / · 1 and Er D .r; r; : : : ; r / still lies below H . Applying the above argument to the cube with the two antipodal vertices Er and 1E we deduce that '[r; 1] ¸ 1 ¡ r . By the choice of s0 and the nonnegativity of f1; : : : ; fn we obtain that '[0; 1] D '[0; s0]C'[s0; r ]C'[r; 1] ¸ s0 C1¡r , and letting r ! s0 yields '[0; 1] ¸ 1, as required. In order to prove Theorem 6.3, we need a slight extension of Theorem 8.2, where one of the splinters is allowed to lie outside the cube. Proposition 8.3. Suppose n ¸ 3. Let a1; : : : ; an be nonnegative real numbers, at most one of which exceeds 1, and let pj D aj ej . Then the set S D convfp1; : : : ; pn; 1Eg \ Qn Proof. We may assume that aj · 1 for all j 6D 1. Let k be the least nonnegative integer such that a1 · ..n ¡ 1/=.n ¡ 2//k . We proceed by induction on k. The case k D 0 was handled in Theorem 8.2, so we assume that k ¸ 1, i.e., a1 > 1. Consider the box B0 D QnjD1[0; bj ], where b1 D 1 and for 2 · j · n we have bj D qaj , with q D .2 ¡ 1=a1/=.n ¡ 1/. It is straightforward to check that the hyperplane determined by p1; : : : ; pn passes through the center of B0 and meets all its facets. It follows by Theorem 5.4 that S is B0-hefty. The box B0, together with the boxes Bj D Plj .[bj ; 1]/ for 2 · j · n, forms a ¿ ¤-determining family of sub-boxes of Qn. (Note that because we set b1 D 1, the box B1 has vanished, but the argument showing that the family is ¿ ¤-determining remains valid.) Thus, it suffices to show that S is Bj -hefty for 2 · j · n. We show this for j D n, for example. The set S contains the set S0 D convfp01; : : : ; p0n; 1Eg \ Bn, where for 1 · i · n ¡ 1 and pi0 D .1 ¡ q/pi C qpn p0n D pn: Now, S0 relates to Bn in the same way as S relates to Qn in the statement of the proposition (up to a normalization in the nth direction). Moreover, denoting by a10 the first coordinate of p01, we have a10 D .1 ¡ q/a1 < n ¡ 2 a1 · : n ¡ 1 n ¡ 2 Hence, by the induction hypothesis, the set S0 (and therefore S) is Bn-hefty. µ n ¡ 1 ¶k¡1 8.2. Proof of the General Case of Theorem 6.3 We may assume that n ¸ 3, and that the set S in question is of the form S D convfp1; : : : ; pn; q1; : : : ; qng, where pj D aj ej and qj D 1E ¡ bj ej for some numbers aj ; bj between 0 and 1. Let F D . f1; : : : ; fn/ be a system of nonnegative-valued functions covering S. We have to prove that P fj [0; 1] ¸ 1, or equivalently, using the notation '.s/ D Pj fj .s/, that '[0; 1] ¸ 1. This is trivially true if fs: '.s/ · 1g D ;, so we assume that this set is nonempty, and consider: s0 D inffs: '.s/ · 1g; Clearly, '[0; s¤]C'[1¡s¤; 1] ¸ 2s¤, and it remains to prove that '[s¤; 1¡s¤] ¸ 1¡2s¤. We consider the two points sE¤ and 1¡¡¡¡!s¤ on the diagonal of Qn, and the cube B¤ having these two points as antipodal vertices. We denote by H0 (respectively H1) the hyperplane determined by p1; : : : ; pn (respectively q1; : : : ; qn), and by I the segment of the diagonal between H0 and H1. We distinguish between several cases concerning the position of sE¤ and 1¡¡¡¡!s¤ relative to I . Case 1: Both points lie in I . In this case the diagonal [sE¤; 1¡¡¡¡!s¤] of B¤ is contained in S and hence '[s¤; 1 ¡ s¤] ¸ 1 ¡ 2s¤. Case 2: Exactly one of the points lies in I . We assume without loss of generality that 1¡¡¡¡!s¤ lies in I (and hence in S) and that sE¤ lies below H0. For 1 · i · n, let pi0 be the point on the intersection of H0 with the line parallel to the xi -axis going through sE¤. We check that pi0 is in B¤, that is, its i th coordinate ai0 does not exceed 1 ¡ s¤. Indeed, we have .1=ai / ai0 C Pj6Di .1=aj / s¤ D 1, and hence ai0 D ai .1 ¡ s¤ Pj6Di 1=aj / · 1 ¡ s¤. Thus, we may apply Theorem 8.2 to the cube B¤ and obtain that '[s¤; 1 ¡ s¤] ¸ 1 ¡ 2s¤. Case 3: None of the points lies in I . We assume without loss of generality that s¤ D s0. Let r ¸ s¤ be such that '.r / · 1 and Er; 1¡¡¡!r still lie outside I . For 1 · i · n, let pi0 (respectively qi0 ) be the point on the intersection of H0 (respectively H1) with the line parallel to the xi -axis going through Er (respectively 1¡¡¡!r). As was shown above, the points pi0 all lie in the cube B D [r; 1 ¡ r ]n, and the same holds for the points qi0 . We consider a ¿ ¤-determining family of sub-boxes of B as in Lemma 7.9. It consists of B0 D Qj [r; aj0 ], where aj0 is the j th coordinate of p0j , and of Bk D [ak0 ; 1 ¡ r ] £ Qj6Dk [r; 1 ¡ r ]; 1 · k · n. As was shown in the proof of Theorem 8.2, the fact that '.r / · 1 implies that t .B0; F / ¸ 1. To each of the Bk ’s we may apply Proposition 8.3. Indeed, the set S contains the splinters q01; : : : ; q0n of the vertex 1¡¡¡!r of Bk , as well as the antipodal vertex of Bk , namely p0k . Among the splinters q01; : : : ; q0n, only q0k may lie outside Bk . Thus, we conclude from Proposition 8.3 that S is Bk -hefty for 1 · k · n. It follows that t .B; F / ¸ 1, that is, '[r; 1 ¡ r ] ¸ 1 ¡ 2r . Using the nonnegativity of f1; : : : ; fn and letting r ! s¤, we obtain that '[s¤; 1 ¡ s¤] ¸ 1 ¡ 2s¤. 2 Acknowledgment We thank an anonymous referee for carefully reading the paper and suggesting many improvements in matters of style. 1. R. Aharoni, R. Holzman and M. Krivelevich, On a theorem of Lova´sz on covers in r -partite hypergraphs, Combinatorica 16 (1996), 149–174. 2. R. Alexander, A problem about lines and ovals, Amer. Math. Monthly 75 (1968), 482–487. 3. K. Ball, The plank problem for symmetric bodies, Invent. Math. 104 (1991), 535–543. 4. Th. Bang, On covering by parallel strips, Mat. Tidsskr. B (1950), 49–53. 5. Th. Bang, A solution of the “plank problem,” Proc. Amer. Math. Soc. 2 (1951), 990–993. 6. Th. Bang, Some remarks on the union of convex bodies, in: Proceedings Tolfte Skandinaviska Matematikerkongressen, Lund, 1953, pp. 5–11, Lunds Universitets Matematiska Institution, 1954. 7. C. Berge, Hypergraphs: Combinatorics of Finite Sets, North-Holland, Amsterdam, 1989. 8. 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Aharoni, Holzman, Krivelevich, Meshulam. Fractional Planks, Discrete & Computational Geometry, 2002, 585-602, DOI: 10.1007/s00454-001-0088-x