Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 39 (1998), No. 2, 473-479.

The Minimum Mean Width Translation Cover for Sets of Diameter One

Károly Bezdek* Robert Connelly

Eötvös University, Institute of Mathematics II, Department of Geometry H-1088 Budapest, Rákóczi út 5, Hungary e-mail: kbezdek@ludens.elte.hu kbezdek@math.cornell.edu Cornell University, Department of Mathematics White Hall, Ithaca, NY 14853-7901, USA e-mail: connelly@math.cornell.edu

Abstract. In this note we prove that the minimum mean width translation cover for all sets of diameter one in E^{d} is the d-dimensional ball of diameter \sqrt{\frac{2d}{d+1}}. This generalizes a well-known theorem of Jung [3].

0. Introduction

Let K be a compact convex set with nonempty interior in d-dimensional Euclidean space E^{d}. In other words, K is a convex body in E^{d}. The support function h_{K} of K is defined by h_{K}(u) = sup{⟨x, u⟩ | x ∈ K} for any vector u in E^{d}, where ⟨x, u⟩ denotes the standard inner product of the vectors x and u. Let S^{d−1} be the unit sphere in E^{d} centered at the origin. That is, S^{d−1} = {x ∈ E^{d} | ⟨x, x⟩ = 1}. The number w_{K}(u) = h_{K}(u) + h_{K}(−u) for u ∈ S^{d−1} is called the width of K in the direction u. Finally, the mean value of the width function over S^{d−1} is called the mean width w(K) of K. Thus,

w(K) = \frac{2}{ω_{d}} ∫_{S^{d−1}} h_{K}(u)dL^{d−1}(u),

where L^{d−1} denotes the (d−1)-dimensional spherical Lebesgue measure computed in S^{d−1} and

ω_{d} = L^{d−1}(S^{d−1}) = \frac{2π^{\frac{d}{2}}}{Γ(\frac{d}{2})}

* The author was partially supported by the Hungarian National Science Foundation, grant no. A-221/95 and by the Geometry Project of the Research Developments Foundation of the Hungarian Ministry of Culture and Education, grant no. FKFP0152/1997.

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix}1998 Heldermann Verlag

474 K. Bezdek, R. Connelly: The Minimum Mean Width Translation Cover …

is the surface area of S^{d−1}, and Γ is the gamma function.

Now, let F be a family of sets in E^{d}. We say that the convex body C of E^{d} is a translation cover of F if each member of F lies in some translate of C. Recall that Jung’s theorem ([3], p.84–85) asserts that any set of diameter one in E^{d} can be covered by (a

translate of) a d-dimensional ball of diameter \sqrt{\frac{2d}{d+1}}. In this note we prove the following stronger statement.

Theorem. The unique minimum mean width translation cover for all sets of diameter one in E^{d} is the d-dimensional ball of diameter \sqrt{\frac{2d}{d+1}}.

The proof of this Theorem presented in Sections 1, 2 and 3 gives the following characterization of balls. (See also the Lemma in Section 3.)

Remark 1. The d-dimensional ball is the only convex body in E^{d} such that all circumscribed tangential regular d-dimensional simplices are congruent and meet the convex body at the centers of their facets.

If K is a convex domain in E^{2}, then the perimeter of K is equal to π · w(K). Thus, the following remark is an immediate corollary of the Theorem. (This corollary has been claimed in [1] without proof. A proof based on the proper Fourier expansion of the support function can be found in [2]. The proof in Sections 1 and 2 is an even shorter one.)

Remark 2. The smallest perimeter convex domain that is a translation cover for all planar sets of diameter 1 is the circle of diameter \frac{2\sqrt{3}}{3}.

Recall that the Reuleaux triangle of constant width 1 is defined as the intersection of the three disks of radius 1 centered at the vertices of a regular triangle of side length 1.

Figure 1

Notice that the Reuleaux triangle of constant width 1 can be placed in a square Q of side length 1, and can then be turned freely within Q while maintaining contact with all four sides of Q. The covered part of Q has area \frac{π}{6} + 2\sqrt{3} − 3 = 0.9877 …. It is not hard to show that the above truncated unit square is a translation cover for all planar sets of diameter one (see for example [1]). Thus, the following problem, that has been raised independently by B. C. Rennie (1977) [4] and by K. Bezdek and R. Connelly (1989) [1] several years ago, is a rather natural question. Surprisingly, it is still open (Figure 1).

Problem. Prove or disprove that the smallest area translation cover for all planar sets of diameter one is the truncated unit square of area \frac{π}{6} + 2\sqrt{3} − 3 = 0.9877 ….

For related results on translation covers see [1].

1. A lower bound for the mean width of a translation cover in E^{d}

Jung’s theorem [3] implies that the lower bound stated in the following claim is in fact a sharp one. (In fact, this is our Theorem without the uniqueness condition.)

Proposition. If C is an arbitrary translation cover for all sets of diameter 1 in E^{d}, then the mean width w(C) ≥ \sqrt{\frac{2d}{d+1}}.

Proof. Let V be an arbitrary d-dimensional regular simplex of edge length 1 centered at the origin o in E^{d}. Let v_{1}, v_{2}, …, v_{d+1} be the vertices of V. Moreover, let u_{i} = \frac{1}{∥v_{i}∥}v_{i} with ∥v_{i}∥ = \sqrt{⟨v_{i}, v_{i}⟩} for all 1 ≤ i ≤ d + 1. Notice that

∑_{i=1}^{d+1} u_{i} = 0

and ⟨v_{i}, u_{i}⟩ = \sqrt{\frac{d}{2(d+1)}} for all 1 ≤ i ≤ d + 1. It is easy to see that the Proposition follows from the inequality

∑_{i=1}^{d+1} w_{C}(u_{i}) ≥ (d + 1)\sqrt{\frac{2d}{d + 1}}.

Finally, the inequality in question can be proved as follows.

Without loss of generality we may assume that V is a subset of C. Moreover, as C has a translate that contains −V = {−x ∈ E^{d} | x ∈ V }, there exists a (translation) vector t such that the vertices t − v_{1}, t − v_{2}, …, t − v_{d+1} of the d-dimensional regular simplex V^{∗} = t + (−V) of edge length 1 all belong to C. Now, let K = V ∪ V^{∗}. Since K ⊂ C we get that

∑_{i=1}^{d+1} w_{C}(u_{i}) ≥ ∑_{i=1}^{d+1} w_{K}(u_{i}) = ∑_{i=1}^{d+1} h_{K}(u_{i}) + h_{K}(−u_{i}) ≥ ∑_{i=1}^{d+1}⟨v_{i}, u_{i}⟩ + ⟨t − v_{i}, −u_{i}⟩

= 2 ∑_{i=1}^{d+1}⟨v_{i}, u_{i}⟩ − ⟨t, ∑_{i=1}^{d+1} u_{i}⟩ = 2(d + 1)\sqrt{\frac{d}{2(d + 1)}} = (d + 1)\sqrt{\frac{2d}{d + 1}}.

This completes the proof of the Proposition. □

The above proof leads to the following important observation.

Corollary. If C is a minimal mean width translation cover for all sets of diameter 1 in E^{d}, then any d-dimensional regular simplex of edge length 1 in E^{d} has a translate with vertices say, v_{1}, v_{2}, …, v_{d+1} such that each v_{i} is a boundary point of C. Moreover, the hyperplane through the vertex v_{i} parallel to the facet of the d-simplex opposite to v_{i} is a supporting hyperplane of C at v_{i}, 1 ≤ i ≤ d + 1.

Remark 3. The existence of a minimal mean width translation cover for all sets of diameter 1 in E^{d} can be proved using standard arguments (see for example [1]).

2. The smallest mean width translation cover in E^{2}: the proof of the Theorem in E^{2}

Let C be a smallest mean width translation cover for all sets of diameter 1 in E^{2}. Without loss of generality we may assume that the origin o of E^{2} is an interior point of the convex domain C. Also, assume that a Cartesian (x, y)-coordinate system is given. Then there is a one-to-one correspondence (modulo 2π) between the vectors u ∈ S^{1} and the angle, say θ, between u and the positive x-axis; in other words: u = u(θ) = (cos θ, sin θ). Consequently, we write the restricted support function h_{C} of C (defined on S^{1}) as a periodic function h_{C}(θ) of the angle θ.

Now, let v_{1}^{∗}, v_{2}^{∗}, v_{3}^{∗} be the vertices of a positively oriented regular triangle of side length 1 in E^{2} such that the side v_{2}^{∗}v_{3}^{∗} is parallel to the y-axis. Then the Corollary implies that there exists a translate Δv_{1}v_{2}v_{3} of the Δv_{1}^{∗}v_{2}^{∗}v_{3}^{∗} such that the vertices v_{1}, v_{2}, v_{3} are all boundary points of C. Moreover, the line ℓ_{i} parallel to the side v_{i−1}v_{i+1} and passing through v_{i} is a supporting line of C at v_{i}, where i − 1, i, i + 1 are computed in mod 3, 1 ≤ i ≤ 3 (Figure 2). Finally, let w_{i} = ℓ_{i−1} ∩ ℓ_{i+1}, 1 ≤ i ≤ 3. Assume that v_{1} is a regular boundary point of C that is assume that h′_{C}(0) exists. Notice that if the coordinates of w_{1} are x_{1} and y_{1}, then y_{1} is the y-coordinate of v_{1} as well and so y_{1} = h′_{C}(0). Moreover, −\frac{1}{2}x + \frac{\sqrt{3}}{2}y = h_{C}(\frac{2π}{3} ) (resp., −\frac{1}{2}x − \frac{\sqrt{3}}{2}y = h_{C}(−\frac{2π}{3} )) is the standard distance equation of the line ℓ_{2} (resp., ℓ_{3}). As w_{1} = ℓ_{2} ∩ ℓ_{3} we have that −\frac{1}{2}x_{1} + \frac{\sqrt{3}}{2}y_{1} = h_{C}(\frac{2π}{3} ) and −\frac{1}{2}x_{1} − \frac{\sqrt{3}}{2}y_{1} = h_{C}(−\frac{2π}{3} ). Hence, \sqrt{3}y_{1} = h_{C}(\frac{2π}{3} )−h_{C}(−\frac{2π}{3} ). Recall that y_{1} = h′_{C}(0). Thus, h′_{C}(0) = \frac{1}{\sqrt{3}}[h_{C}(\frac{2π}{3} ) − h_{C}(−\frac{2π}{3} )]. Obviously, this implies that h′_{C}(θ) = \frac{1}{\sqrt{3}}[h_{C}(θ + \frac{2π}{3} ) − h_{C}(θ − \frac{2π}{3} )] for all θ for which h′_{C}(θ) exists. As h_{C}(θ) is absolutely continuous we get that h′_{C}(θ) is absolutely continuous as well and h′′_{C}(θ) = \frac{1}{\sqrt{3}}[h′_{C}(θ + \frac{2π}{3} ) − h′_{C}(θ − \frac{2π}{3} )] for almost all θ. Thus, h′′_{C}(θ + \frac{2π}{3} ) + h_{C}(θ + \frac{2π}{3} ) = \frac{1}{\sqrt{3}}[h′_{C}(θ + \frac{4π}{3} )−h′_{C}(θ)]+h_{C}(θ+ \frac{2π}{3} ) = \frac{1}{3}[h_{C}(θ)−h_{C}(θ+ \frac{2π}{3} )−h_{C}(θ+ \frac{2π}{3} )+h_{C}(θ− \frac{2π}{3} )]+h_{C}(θ+ \frac{2π}{3} ) = \frac{1}{3}[h_{C}(θ− \frac{2π}{3} )+h_{C}(θ)+h_{C}(θ+ \frac{2π}{3} )] = \frac{1}{3} \sqrt{3} = \frac{1}{\sqrt{3}} for almost all θ. Hence, h′′_{C}(θ)+h_{C}(θ) = \frac{1}{\sqrt{3}} for almost all θ. Thus h_{C}(θ) must be equal to the general solution of the above secondorder nonhomogeneous linear differential equation that is h_{C}(θ) = \frac{1}{\sqrt{3}} + a cos θ + b sin θ. This is the support function for a circle whose center is at (a, b) with radius \frac{1}{\sqrt{3}}. This completes the proof of the Theorem in E^{2}.

Figure 2

3. The smallest mean width translation cover in E^{d}: the proof of the Theorem in E^{d}

The Corollary implies that in order to prove the Theorem it is sufficient to show the following.

Lemma. If C is a convex body in E^{d} with the property that any d-dimensional regular simplex of edge length 1 in E^{d} has a translate with vertices say, v_{1}, v_{2}, …, v_{d+1} such that each v_{i} is a boundary point of C, and the hyperplane through the vertex v_{i} parallel to the facet of the d-simplex opposite to v_{i} is a supporting hyperplane of C at v_{i}, 1 ≤ i ≤ d + 1, then C is a d-dimensional ball of diameter \sqrt{\frac{2d}{d+1}}.

Proof. We prove the Lemma by induction on the dimension d. Since the claim has been proved for d = 2, we assume that it is true in any d′-dimensional Euclidean space with d′ ≤ d − 1 and we prove it in E^{d}.

Let C be a convex body in E^{d} with the property described in the Lemma. Let u ∈ S^{d−1} be a unit vector in E^{d} and let H_{u} (resp., H_{u}^{+}) be the supporting hyperplane (resp., supporting closed halfspace) of C with outer normal vector u. (So C ∩ H_{u} ≠ ∅, H_{u} ∩ H_{u}^{+} = H_{u} and C ⊂ H_{u}^{+}.) Now, take any d-dimensional regular simplex of edge length 1 in E^{d} with vertices v_{1}^{∗}, v_{2}^{∗}, …, v_{d+1}^{∗} such that v_{1}^{∗} ∈ H_{u}, {v_{2}^{∗}, …, v_{d+1}^{∗}} ⊂ H_{u}^{+} and the facet determined by v_{2}^{∗}, …, v_{d+1}^{∗} is parallel to H_{u}. Then this has a translate that

Figure 3

is a regular d-simplex of edge length 1 with vertices v_{1}, v_{2}, …, v_{d+1} such that v_{1} ∈ H_{u}, {v_{2}, …, v_{d+1}} ⊂ H_{u}^{+} and the hyperplane H_{u}^{∗} spanned by the vertices v_{2}, …, v_{d+1} is parallel to H_{u} and all of the vertices v_{1}, v_{2}, …, v_{d+1} lie on the boundary of C. Thus, the

induction hypothesis implies that H_{u}^{∗} ∩C is a (d−1)-dimensional ball of diameter \sqrt{\frac{2(d−1)}{d}}. Moreover, v_{1} is fixed for a given u that is we can write v_{1} as the corresponding support point s(u) of C. Also, if we take the corresponding support hyperplanes of C through the vertices v_{2}, …, v_{d+1}, then they all intersect in a single point that depends on u only. We call this point c(u) the cone point of C assigned to u. (See Figure 3). Obviously, the line ℓ(u) connecting the points s(u) and c(u) is orthogonal to H_{u}^{∗} and passes through the center of the (d − 1)-ball H_{u}^{∗} ∩ C.

Finally, notice that if u_{1} ∈ S^{d−1} and u_{2} ∈ S^{d−1} are sufficiently close to each other, then the congruent (d − 2)-dimensional spheres H_{u_{1}}^{∗}∩ bd C and H_{u_{2}}^{∗}∩ bd C intersect each other, where bd C denotes the boundary of C. In fact, the points in common are the boundary points of C that lie on supporting hyperplanes of C passing through the cone points c(u_{1}) and c(u_{2}). Suppose that v is such a point. Then the simplices with unit edge length, guaranteed by the hypothesis, that are in contact with K at v all share the same centroid, which must also lie on ℓ(u_{1}) and ℓ(u_{2}). Thus, ℓ(u_{1}) ∩ ℓ(u_{2}) ≠ ∅ for any two sufficiently close unit vectors u_{1} and u_{2} in S^{d−1}. Hence, ∩{ℓ(u) | u ∈ S^{d−1}} ≠ ∅ and C is

the ball of diameter \sqrt{\frac{2d}{d+1}}. This completes the inductive proof of the Lemma. □

References

[1] Bezdek, K.; Connelly, R.: Covering curves by translates of a convex set. Amer. Math. Monthly 96 (1989), 789–806.

[2] Bezdek, K.; Connelly, R.: Minimal translation covers for sets of diameter 1. Preprint (1997), 1–5.

[3] Bonnesen, T.; Fenchel, W.: Theorie der konvexen Körper. Springer, Berlin 1934; English translation: BCS Associates, Moscow, Idaho, 1987.

[4] Klee, V.; Wagon, S.: Old and new unsolved problems in plane geometry and number theory. The Dolciani Mathematical Expositions, MAA (1991).

Received August 19, 1997