Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 39 (1998), No. 2, 259-262.

Covering a Three-Dimensional Convex Body by Smaller Homothetic Copies

Marek Lassak

Instytut Matematyki i Fizyki ATR ul. Kaliskiego 7, 85-796 Bydgoszcz, Poland e-mail: lassak@atr.bydgoszcz.pl

Abstract. We show that every three-dimensional convex body C can be covered by 20 smaller positive homothetic copies. The homothety ratio here depends on C. Under the requirement that the homothety ratio is universal, such a covering is possible by 24 copies. MSC 1991: 52C17

Let C be a convex body in Euclidean d-space E^{d}. Following [4], we denote by L(C) the smallest m such that C can be covered by m homothetic copies of a positive ratio smaller than 1. The well-known conjecture of Hadwiger says that L(C) ≤ 2^{d} for every convex body C ⊂ E^{d} (see [5]). From the example of the parallelotope it follows that the number 2^{d} cannot be lessened. The problem remains open for d ≥ 3. It is solved only for some special classes of convex bodies in E^{d}. The reader can find details in the surveys of results concerning Hadwiger’s problem in [2] and [3].

From the d-dimensional estimate in Corollary 4 from [8] we see that L(C) ≤ 34 for every three-dimensional convex body C. The aim of this paper is to improve this estimate up to L(C) ≤ 20 for every convex body C ⊂ E^{3}. Recall that if C is centrally-symmetric, then L(C) ≤ 8 (see [7]), and that if C is of constant width, then L(C) ≤ 6 (see [9]).

Let l be a direction and let δ > 0. Denote by G_{l}(C) the union of all straight lines parallel to l which intersect a convex body C but have empty intersection with the interior of C. By the δ-tube of C with respect to l we mean the set of points of conv(G_{l}(C)) which are in the distance smaller than δ from G_{l}(C). The symbol “conv” here means the operation of creating the convex hull. By the part of C inside of the δ-tube of C with respect to l we mean the set of points of C which do not belong to the δ-tube itself.

Lemma. Let C ⊂ E^{d} be a convex body. For every direction l and every δ > 0 there are two smaller homothetic copies of C which cover the part of C inside of the δ-tube of C with respect to l.

260 M. Lassak: Covering a Three-Dimensional Convex Body …

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix}1998 Heldermann Verlag Proof. Denote by W the part of C inside of the δ-tube of C with respect to l. If W = ∅, our Lemma is obvious. Assume that W ≠ ∅. Of course, W can be presented as the union of non-empty vertical segments lying in different vertical straight lines. Since W is a closed set, there is a shortest from those segments and its length η is greater than 0. We see that W is contained in the union of the interiors of the two translates C_{1} and C_{2} of C by \frac{1}{3}η units in the directions l and −l. Since W is a closed set, it is also contained in the union of some smaller positive homothetic images C′_{1} and C′_{2} of C_{1} and of C_{2}, respectively. Of course, C′_{1} and C′_{2} are smaller positive homothetic copies of C.

Theorem. Every three-dimensional convex body can be covered by 20 smaller homothetic copies.

Proof. Without loss of generality, we may assume that the diameter of our body C is 1. Let a and b be points of C at the distance 1. Assume that the segment ab is vertical with a above b. Construct the horizontal planes H_{a} and H_{b} through a and b, respectively. Of course, the planes support C.

Denote by P_{a} and P_{b} the vertical projections of C onto H_{a} and H_{b}. Observe that C is contained in the cylinder Q = conv(P_{a} ∪ P_{b}). Let K be the set of points of C whose projections on H_{b} belong to the relative boundary of P_{b}. We can present K as the union of vertical segments (including one-point segments) lying in different vertical straight lines. We take the highest point from every such vertical segment. The set of those highest points is denoted by M. We support M from the top and from the bottom by horizontal planes and we denote by ω the distance between them. We have ω < 1 because otherwise the diameter of C would be greater than 1.

Put τ = \frac{1}{3+ω} + \frac{1}{4}. Since 0 ≤ ω < 1, we have \frac{1}{2} < τ. Let μ = \frac{5−ω}{4} . From ω < 1, we obtain μ > 1. We choose λ < 1 such that λτ ≥ \frac{1}{2} and λμ ≥ 1. Take homothetic copies M_{a} and M_{b} of M with the homothety ratio τ and with homothety centers a and b, respectively. Put D = conv(M_{a} ∪ M_{b}). Observe that D ⊂ C and that D is the union of vertical segments of length at least 1 − τ.

Denote by j a unit vector parallel to the direction l of the segment ab, and by v an arbitrary vector. The union of the three translates of λD by vectors v, v + λ(1 − τ)j and v + 2λ(1 − τ)j contains a right cylinder R(v) of height λ[3(1 − τ) − ωτ] = λ\frac{5−ω}{4} = λμ whose base is a translate of λτP_{b}.

Levi [10] proved that every plane convex body A can be covered by 7 homothetic copies of ratio \frac{1}{2}. This covering is obtained thanks to the well-known result that an affineregular hexagon can be inscribed in A (see [1]). Homothety centers of 6 of them are in the vertices of the two triangles whose common part is the hexagon, and one in the center of the hexagon. The 6 copies cover the boundary of A together with a neighbourhood of it. One copy is in the interior of A.

We apply the above covering by 7 copies of ratio \frac{1}{2} to P_{b} as A. Since λτ ≥ \frac{1}{2}, we can also cover P_{b} by 7 homothetic copies of τP_{b} of ratio λ. Consider the 6 copies A_{1}, …, A_{6} which cover the boundary of P_{b}. Over every such copy A_{i}, we put 3 translates of λD in such a way that the base of the corresponding cylinder R(v) is just A_{i}. Since the height λμ of R(v) is at least 1, we see that there is a δ > 0 such that the 18 translates of λD cover the part of Q which is a subset of the δ-tube of C of direction l. Since C ⊂ Q, they cover the part of C which is contained in the δ-tube of C of direction l. Since every translate of λD is a subset of a homothetic copy of C of ratio λ, we also obtain a covering by 18 smaller homothetic copies of C. Thanks to the Lemma, two additional smaller homothetic copies of C cover the part of C inside of the δ-tube of C. We see now that 20 smaller homothetic copies of C cover C.

A sketch of this proof was presented by the author in his talk during the conference “Convex Bodies” in Oberwolfach in 1984, comp. [2, p. 270], and [11, p. 389].

Since ω in the proof of the Theorem cannot be bounded from above by a constant smaller than 1, our τ and λ may be arbitrarily close to 1. Thus in the Theorem we do not establish a universal homothety ratio of our 20 homothetic copies. Of course, a fixed ratio smaller than 1 could be given for every particular convex body. The following Proposition guarantees a covering by homothetic copies with a fixed universal ratio smaller than 1. As a price, the number of the homothetic copies increases.

Proposition. Every three-dimensional convex body can be covered by 24 homothetic copies of ratio \frac{7}{12} \sqrt{2} + \frac{1}{6} = 0.9916 …

Proof. We apply the fact that every planar convex body can be covered by 4 homothetic copies of ratio \frac{1}{2} \sqrt{2} (see [6]). The approach is similar to this in the proof of the Theorem. The difference is that now we put vertically 6 copies of λD in 4 columns and that we take τ = 6/(7 + \sqrt{2}) and λ = \frac{7}{12} \sqrt{2} + \frac{1}{6} = 0.9916 … We have λ\sqrt{2}τ = λ[6(1 − τ) − τ] = 1.

Similarly we can show that every three-dimensional convex body can be covered by 28 homothetic copies of ratio \frac{7}{8}. We put vertically 4 copies of λD in 7 columns as in the proof of the Theorem and we take τ = \frac{4}{7} and λ = \frac{7}{8}. We have λ2τ = λ[3(1 − τ) − τ] = λ\frac{8}{7} = 1.

References

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[3] Croft, H. T.; Falconer, K. J.; Guy, R. K.: Unsolved Problems in Geometry. Springer, 1991.

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[5] Hadwiger, H.: Ungelöste Probleme. Nr. 20. Elem. Math. 12 (1957), 121.

[6] Lassak, M.: Covering a plane convex body by four homothetical copies with the smallest positive ratio. Geom. Dedicata 21 (1986), 151–167.

[7] Lassak, M.: Solution of Hadwiger’s covering problem for centrally-symmetric convex bodies in E^{3}. J. London Math. Soc. (2) 30 (1984), 501–511.

[8] Lassak, M.: Covering the boundary of a convex set by tiles. Proc. Amer. Math. Soc. 104 (1988), 269–272.

[9] Lassak, M.: Illumination of three-dimensional convex bodies of constant width. Proceedings of 4-th International Congress of Geometry, Thessaloniki 1996, 246–250.

[10] Levi, F. W.: Ein geometrisches Überdeckungsproblem. Arch. Math. 5 (1954), 476– 478.

[11] Martini, H.: Some results and problems about zonotopes. In: Colloq. Math. Soc. János Bolyai 48, “Intuitive Geometry”, Siófok 1985, North Holland, 383–418.

Received July 14, 1997