Ellipsoids are the Most Symmetric Convex Bodies

Peter M.Gruber and Tibor Ódor

Abstract. A smooth convex body is a solid ellipsoid if and only if it= appears centrally symmetric from any exterior point. Œ

1 Introduction and Statement of Results

1.1 The problem

In this article we deal with the following conjectures proposed in= [3, 10]: let C be a convex body in Euclidean= d-space IE^{d}, d ≥ 3, i.e. a compact convex subset of IE^{d} with= non-empty interior. Assume that for each exterior point p the visual= cone of C with vertex p,

C(p, C) = 3D{p + t(x − p) : t ≥ 0, x ∈ C},

is symmetric (with respect to a suitable axis A(p, C)). Then C is a solid= ellipsoid. It is even presumed that it suffices to consider the= points p on a closed hypersurface containing C in its interior.

Dual versions of these conjectures are the following: C is a solid= ellipsoid if each intersection of C with a hyperplane has a = projective center, that is, it has a center after a suitable projective= transformation has been applied to it. It is sufficient to consider only= those hyperplanes which are tangent to a closed hypersurface in the= interior of C.

1.2 Relations to other problems

1.2.1 Geometric tomography. Assume that certain properties of some or= all visual cones of a convex body C are known. What can be said about= C? For pertinent results see Matsuura [17], Bianchi and Gruber= [3]; and Bol [6] and Klee [12]. Compare also the= discussions in Gardner [9], pp. 125, 200, 201.

1.2.2 Caustics of billiards. A (convex) billiard table B is a= closed convex surface in IE^{d}, d ≥ 2, that is the boundary of a convex= body. A billiard ball is a point which moves at constant speed along= a straight line in the interior of B until it hits B. If the hitting= point is a smooth point of B, the billiard ball is reflected in the usual= way and moves again at the same speed along a straight line, etc. If this= point is singular, the ball stops there. The curve described by a billiard= ball is called its trajectory. A compact convex set C in the= interior of B is a (convex) caustic of B if each trajectory which= touches C once, touches it again after each reflection. The latter is= equivalent to the following: for each smooth point p of B the cone= cone(p, C) is symmetric with respect to the interior normal of B at= p. Thus, if the second of the above conjectures were true, it could be= used to show that for d ≥ 3 the only billiard tables with caustics are= the ellipsoids and their caustics are precisely their interior confocal= solid ellipsoids and the intersection of these. The direct proof of this= result is due to Gruber [11], the smooth case was settled before= by Berger [2]. See also the discussions by Klee and Wagon= [13], pp. 75, 76, and Gardner [9], pp. 198, 200. For= d = 3D2 all sufficiently smooth billiard tables with positive curvature= have ‘many’ caustics as shown by Lazutkin [15] and Douady= [8], while ‘most’ billiard tables in the sense of Baire= categories have no caustics, see Gruber [10]. Caustics are= related to eigenfunctions of the Laplace operator, compare the references= in Lazutkin [15].

1.2.3 Brunn’s characterization of ellipsoids. A well-known result of= Brunn [7] says that a convex body in IE^{d}, d ≥ 3, is a solid= ellipsoid if and only if each of its hyperplane sections has a center.= There are two natural ways to refine Brunn’s theorem: first, to use fewer= hyperplanes and second, to relax the symmetry assumption. Refinements of= the first type are due to Olovjanischnikov [18], Aitchison, Petty= and Rogers [1], and Larman [14]. For a refinement of= the second type see Theorem 2 below.

1.3 Results

Our contributions to the above conjectures are the following two results:

Theorem 1. Let C be a convex body in IE^{3}, d ≥ 3, (whose= boundary is a surface) of class C^{4} with positive Gauss curvature. If= for all exterior points p of C which are sufficiently close to C the= visual cone C(p, C) is symmetric, then C is a solid ellipsoid.

Theorem 2. Let C be as in Theorem 1. If for all hyperplanes H= which meet C and are sufficiently close to a support plane of C, the= intersection C ∩ H has a projective center, then C is a solid= ellipsoid.

2 Tools

bd and cl stand for boundary (with respect to the affine hull) and for= closure. For notions not defined we refer to [21].

2.1 The first tool is due to Pick [20]; see Blaschke= [4], p. 109.

(1) Let C be a convex body in IE^{3} of class C^{3} with positive Gauss= curvature and let p ∈ bd C. Then after the introduction of a suitable= affine coordinate system in IE^{3} with origin p, one may represent a= neighborhood of p in bd C in the form

z = 3Dx^{2} + y^{2} + c(x^{3} − 3xy^{2}) + o(|x|^{3} + |y|^{3}) as |x| + |y| →= 0,

where c is constant.

2.2 The second tool is a well-known theorem of Maschke [16];= see also Blaschke [4], p. 118, and Paukowitsch [19].

(2) Let F be a connected surface in IE^{3} of class C^{4} with Gauss= curvature ≠ 3D0. If at each point of F a quadric approximates F = of order 3, then F is itself a piece of a quadric.

2.3 R. Schneider informs us about the following results; see also= [21], p. 117.

(3) Let C be a convex body in IE^{d}, d ≥ 3, of class C^{k}, k ≥ 2, with= positive Gauss curvature and let S be a linear subspace of IE^{d} of= codimension 1. Then the orthogonal projection of C into S is a convex= body in S of class C^{k} with positive Gauss curvature.

(4) Let C be as in (3) with the origin in its interior. Then the polar body= C^{∗} = 3D{x : x·y ≤ 1 for all y ∈ C} of C is also of class= C^{k} with positive Gauss curvature.

2.4 The final tool is a result of Blaschke and Hessenberg= [5].

(5) Let C be a convex body in IE^{d}, d ≥ 3, and assume that the orthogonal= projection of C into any linear subspace of IE^{d} of codimension 1 is a= solid ellipsoid. Then C is also a solid ellipsoid.

3 Proof of Theorem 1

The proof is by induction on d.

3.1 The case d = 3D3

3.1.1 The main step is to show the following.

(6) Let p ∈ bd C. Then there is a quadric which approximates bd C at p= of order 3.

Choose an affine coordinate system and represent a neighborhood of p = 3Do= in bd C according to (1). Then for the proof of (6) it is sufficient to= show that

(7) c = 3D0.

This will be proved by contradiction in subsections 3.1.2 and 3.1.3. Thus,= by replacing the first two coordinates by their opposite, if necessary, we= may assume that

(8) c > 0.

3.1.2 In this subsection we use the above affine coordinate system.= Let ε_{i}, i = 3D1, 2, …, be a strictly decreasing real= sequence with limit 0 and let p_{i} = 3D(0, 0, −ε_{i}), where (· =, ·, ·) refers to our affine coordinate system. The visual cone= determined by p_{i} and the paraboloid

z = 3Dx^{2} + y^{2}

intersects the plane z = 3D0 in a solid circle (with respect to our affine= coordinate system). Dilating it by the factor 2ε_{i}^{−1/2}= yields the solid circle

K : x^{2} + y^{2} ≤ 1, z = 3D0.

Next, represent a neighborhood of p = 3Do in bd C in the form

(9) z = 3Dz(x, y) = 3Dx^{2} +y^{2} +c(x^{3} −3xy^{2})+o(|x|^{3} +|y|^{3}) as |x|+|y| →= 0,

see (1). Note that C is strictly convex. Thus the visual cone determined= by p_{i} and C coincides for all sufficiently large i with the visual= cone determined by p_{i} and the surface in (9). Intersecting this cone= with the plane z = 3D0 and dilating the intersection by the factor= 2ε_{i}^{−1/2}, gives a convex disc K_{i}. In the rest of this= subsection two properties of K_{i} will be shown.

The first property says that

(10) (1 − 4cε_{i}^{1/2})K ⊂ K_{i} ⊂ (1 + 4cε_{i}^{1/2})K= for all sufficiently large i.

To see this, represent the coordinates x, y in the form

x = 3Dr cos φ, y = 3Dr sin φ, r ≥ 0, 0 ≤ φ ≤ 2π = .

Then, instead of (9) we have

(11) z = 3Dz(r, φ) = 3Dr^{2} + c cos(3φ) · r^{3} + o(r^{3}) as r → 0.

Since o(·) may be chosen independent of φ, the surface in= (11) is in a suitable neighborhood of p = 3Do located between the two= surfaces

(12) z^{±} = 3Dz^{±}(r, φ) = 3Dr^{2} ± 2cr^{3}.

By choosing this neighborhood smaller, if necessary, we may assume that each= of the two surfaces in (12) is strictly convex. Then for all sufficiently= small ε_{i} and thus for all sufficiently large i the following= hold: the visual cone determined by p_{i} and (the surface defined by)= z^{+} is contained in the visual cone determined by p_{i} and z (which= equals C(p_{i}, C)) which in its turn is contained in the visual cone= determined by p_{i} and z^{−}. Intersecting these cones with the plane= z = 3D0 and dilating the intersections by the factor= 2ε_{i}^{−1/2}, elementary calculus arguments imply (10).

The second needed property is the following.

(13) Let δ > 0 be arbitrarily small but fixed and consider the intervals

I_{1} = 3D[−\frac{π}{6} +δ, \frac{π}{6} −δ], I_{2} = 3D[\frac{π}{= 6}+δ, \frac{3π}{6} −δ], …, I_{6} = 3D[\frac{9π}{6} +δ, = \frac{11π}{6} −δ].

Then for all sufficiently large i the arcs of bd K_{i} corresponding to= φ ∈ I_{1}, I_{3}, I_{5} are inside K and the arcs of bd K_{i}= corresponding to φ ∈ I_{2}, I_{4}, I_{6} are outside K.

Since o(r^{3}) in (11) may be chosen independent of φ and c > 0 by= (8),

z = 3Dz(r, φ) = 3Dr^{2} + c cos(3φ)r^{3} + o(r^{3}) \{ \begin{matrix}> r^{2} for φ ∈ I_{1}, I_{3}, I_{5} \\ < r^{2} for φ ∈ I_{2}, I_{4}, I_{6} \end{matrix} \}

for all sufficiently small r. Considering visual cones, this yields (13). The points

h_{1} = 3D(cos \frac{π}{6} , sin \frac{π}{6} , 0), …, h_{6} = 3D(cos = \frac{11π}{6} , sin \frac{11π}{6} , 0) ∈ bd K

form the vertices of a hexagon which is inscribed into K and regular= (with respect to the Euclidean norm with unit disc K).

3.1.3 In this subsection we will prove (7). Since symmetry of cones is= not an affine invariant, we now use a Cartesian coordinate system with= origin p = 3Do such that

(14) K = 3D\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} ≤ 1, z = 3D0 = (a, b > 0),

(15) q_{i} = 3D2ε_{i}^{−1/2}p_{i} = 3D[αε_{i}^{1/2}, = βε_{i}^{1/2}, −γε_{i}^{1/2}] (γ > 0).

Here [·, ·, ·] refers to the Cartesian coordinate system= and x, y, z denote coordinates. Now order IE^{3} by the z-coordinate.

For the proof that

(16) ) <(A(p, C), vertical line through p_{i})

= 3D) <(A(q_{i}, 2ε_{i}^{−1/2}C), vertical line through= q_{i})=3DO(ε_{i}),

consider a vertical plane through q_{i} containing the axis= A(q_{i}, 2ε_{i}^{−1/2}C). The intersection of this plane with the= part of the cone C(q_{i}, 2ε_{i}^{−1/2}C) in the halfspace z ≤ 0= is a solid triangle. By (15) the height of this triangle is= γε_{i}^{1/2}. The distances of the foot of q_{i} on the side= of the triangle in the plane z = 3D0 from the vertices on this side are= τ + O(ε_{i}^{1/2}) and τ + O(ε_{i}^{1/2}) where= τ is a number between a and b, see (10), (14), and (15). Since= A(q_{i}, 2ε_{i}^{−1/2}C) is the bisector of the angle at vertex= q_{i} of the triangle, elementary trigonometry arguments yield (16).

Let L_{i} be the intersection of the cone C(q_{i}, 2ε_{i}^{−1/2}C)= with the plane through p = 3Do normal to its axis= A(q_{i}, 2ε_{i}^{−1/2}C). Clearly, (17) L_{i} is a convex disc, symmetric in p = 3Do.

Let “^{i}” denote the projection with center q_{i} onto the plane= containing L_{i}. Clearly K_{i}^{i} = 3DL_{i}. Using (14), (15), and (16)= together with elementary trigonometry, we see that

(18) K^{i} → K in the Hausdorff metric as i → ∞,

h_{1}^{i} → h_{1}, …, h_{6}^{i} → h_{6} as i → ∞.

Since K_{i}^{i} = 3DL_{i}, propositions (13) and (18) imply that

(19) for all sufficiently large i the following hold: the hexagon with vertices= h_{1}^{i}, …, h_{6}^{i} is inscribed into the solid ellipse K^{i} and is= arbitrarily close to a regular hexagon. The intersection points of bd = K^{i} and bd L^{i} (= 3D bd K_{i}^{i}) are arbitrarily close to= h_{1}^{i}, …, and h_{6}^{i}, respectively. The arcs of bd L_{i} determined= by a pair of successive intersection points which are close to a pair of= successive points among h_{1}^{i}, …, h_{6}^{i}, are alternatingly inside and= outside K^{i}.

Considering the mutual positions of L_{i} and the solid ellipse K^{i} as= described in (19), we may choose three support strips of L_{i}, i.e.p̄arallel strips, bounded by two distinct parallel support lines of L_{i},= such that the middle lines of these strips have no common point. Since this= clearly contradicts (17) the proof of (7) and thus of (6) is complete.

3.1.4 Having proved (6), Maschke’s result (2) shows that bd C is a= piece of a quadric. Since bd C is closed surface, it thus is an= ellipsoid, concluding the proof of Theorem 1 for d = 3D3.

3.2 The case d > 3

We have shown that Theorem 1 holds for d = 3D3. Assume now that d > 3 and= that it holds for d − 1. If S is a linear subspace of codimension 1, let= “^{S}” denote the orthogonal projection of IE^{d} onto S.

3.2.1 Our first proposition is an immediate consequence of (3) and the= assumption of Theorem 1.

(20) Let S be a linear subspace of IE^{d} of codimension 1. Then C^{S} is a= convex body in S of class C^{4} with positive Gauss curvature.

The next step is to show the following.

(21) Let S be as in (20). Then for all exterior points q of C^{S} in a= suitable neighborhood of C^{S} in S the visual cone C(q, C^{S}) is= symmetric.

For the proof of (21) it is sufficient to prove the following claim.

(22) Let S be as in (20) and let p^{S} ∈ bd C^{S}. Then there is a neighborhood= W of p^{S} in S such that for each q ∈ W \C^{S} the visual= cone C(q, C^{S}) is symmetric.

Choose p ∈ bd C with projection p^{S} and let L be an oriented line= through p orthogonal to S. Then we can speak of ‘above’ and ‘= below’ and it is clear what is meant when we say that a ray ‘points= above or below the horizon’. Let U be a convex open neighborhood of= C such that for each q ∈ U\C the visual cone C(q, C) is= symmetric. Choose points u, v ∈ L ∩ U above and below p,= respectively. Since C is strictly convex, p is the only point of C on= L. Thus the cone C(u, C) contains the vertical ray starting at u and= pointing down, but not the opposite ray. Hence its axis A(u, C) points= below the horizon. Now note that (in the natural sense) A(·, C) is= continuous as a function on U\C. We thus may choose a convex= neighborhood u + V ⊂ U\C of u such that for each r ∈= u+V we have that A(r, C) points below the horizon. By choosing V = smaller, if necessary, we may assume that similarly, v + V ⊂= U\C and for each t ∈ v + W we have that A(t, C) points above= the horizon. Now, consider the neighborhood W = 3D(u+V)^{S} = 3D(v +V)^{S} of= p^{S}(= 3Du^{S} = 3Dv^{S}) in S and let q ∈ W \C^{S}. Choose r ∈= u + W and t ∈ v + W with q = 3Dr^{S} = 3Dt^{S}. By continuity, there is a point= s on the line segment r, t and in U\C such that the visual= cone C(s, C) has horizontal axis. Projecting into S then shows that= C(q, C^{S}) = 3D(C(s, C))^{S} has axis (A(s, C))^{S} and thus is symmetric. This= concludes the proof of (22) and thus of (21).

3.2.2 (20) and (21) together with the induction hypothesis imply that= the orthogonal projection of C into any subspace of codimension 1 is a= solid ellipsoid. Since d > 3, this combined with the result (5) of Blaschke= and Hessenberg shows that also C is a solid ellipsoid. Thus Theorem 1= holds for dimension d too. □

4 Proof of Theorem 2

Theorem 2 follows from (4) and Theorem 1 by polarity. □

Acknowledgement

For helpful comments we are obliged to Professor Schnitzer.

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Tibor Ódor

Mathematical Institute

Hungar. Acad. Sci.

Reáltanoda u. 13 – 15

H–1053 Budapest, Hungary odor@math-inst.hu Peter M.Gruber Abteilung für Analysis Techn. Univ. Wien Wiedner Hauptstraße 8 – 10/1142 A–1040 Vienna, Austria pmgruber@pop.tuwien.ac.at