Sections of Bodies of Constant Width

Abstract

In Chapter 3 it was proven that the property of constant width is inherited under orthogonal projection but not under sections. The proof of this fact was not a constructive one, that is, no nonconstant width section of a body of constant width was actually exhibited. In fact, it was proven that if all sections of a convex body have constant width, then the body is a ball. Since there are bodies of constant width other than the ball, it was concluded that they must all have at least one section that is not of constant width. To show this could, however, be tricky, even in cases as simple as the body produced by rotating the Reuleaux triangle around one of its axes of symmetry.

Mighty is geometry; joined with art, resistless.

Euripides

Appendices

Notes

In [377], it is proved that the lengths of the segments, intercepted by a body of constant width h on each of two parallel lines of distance at most d apart, differ by at most $2(2dh)^{\frac{1}{2}}$. This value depends on h, but not on the shape of that body. For parallel sections in higher dimensions, analogous results are obtained.

In [847] the following is investigated: If K is a compact subset of ${\mathbb E}^{n+k}$, what can we recognize about K from some of its n-dimensional linear sections. For example, topological conditions on a family X of n-plane sections are established which assure that K is a body of constant width if the size of X satisfies this topological condition and every section in X has constant width, see also Section 16.4. And it is shown in [845] that sections of convex bodies, which are themselves of constant width, can be used to characterize balls (see Section 9.1 and also [1168] for $n=3$). Inspired by the paper [531], in [59] the following was shown: A convex body in a normed plane is of constant width if the following holds for any chord of it: one of the two parts, into which it is split by this chord, has diameter equal to the length of the chord. The only if part also extends to higher dimensions. Similarly, it was proved in [54] that if a body of constant width in a normed space is cut by a hyperplane, then one of the two parts has the same diameter as the original body. For two dimensions, this property even characterizes sets of constant width. A similar result was proved in [57]. Furthermore, if a body of constant width in a normed space is partitioned into a continuous family of hyperplane sections $H(t), 0 \le t \le 1$, then the diameter of H(t) is a unimodal function of t, and again in two (but not in higher) dimensions this property characterizes bodies of constant width (see again [54]). Constructing suitable bodies of constant width in 3-space, Danzer [276] answered a problem of Süss. Namely, he showed that there are 3-dimensional convex bodies whose own thickness (i.e., minimal width) is larger than the maximum thickness of all their plane sections, see Section 9.1. Also, in Armstrong’s paper [37] planar sections of surfaces of constant width play a role.

Results on sections and section functions related to spherical harmonics, Fourier series, and geometric tomography (see [461],  [455], and [401]) will mainly be discussed in Chapter 13.

Congruent Sections and Projections

Let $\phi$ be 3-dimensional convex body with the following property: to every direction u continuously corresponds a plane $H_u$ orthogonal to u such that all the sections $H_u\cap \phi$ are congruent. By means of Brouwer’s fixed point theorem, Süss [1109] proved in 1947 that all these sections are disks; consequently, $\phi$ is a ball. By using spherical harmonics (see Theorem 13.1.7) it is possible to prove that a centrally symmetric convex n-dimensional body $\phi$ with the property that all its hyperplane sections through the center of symmetry are congruent is a ball. In 1979 Schneider [1037] proved this result without the hypothesis of central symmetry, and in 1990 Montejano [846] proved that if all the sections of a convex body through a point $p_0$ are similar, then the convex body is also a ball, but the point $p_0$ is not necessarily the center of the ball. For analogous problems and results referring to projections, see [848] and [675]. Consider the following related problem: Suppose $\phi$ is a convex $(n+1)$-dimensional body with the property that all its sections through a fixed point $p_0\in$ int$\phi$ are affinely equivalent. Is $\phi$ an ellipsoid? This problem is essentially reflected by the following conjecture of Banach. If $1<k<n$ and $V^n$ is an n-dimensional (real or complex) Banach space with unit ball $\mathcal{G}$ and all the k-dimensional subspaces of $V^n$ are isometric (all the k-sections of $\mathcal{G}$ are affinely equivalent), then $V^n$ is an inner product space ($\mathcal{G}$ is an ellipsoid). Gromov proved in [466] that the conjecture is true if one of the following conditions holds: (1) k is even; (2) k is odd, $V^n$ is complex, $n\ge 2k$; (3) k is odd, $V^n$ is real and $n\ge k+2$. Dvoretzky [305] derived the same conclusion under the hypothesis $n=\infty$. Furthermore, if additionally between any two sections there is an affine volume preserving isomorphism, then the conjecture is true [846]. Lately, some progress has been done by Montejano et al., in the solution of the problem when $k=5,9$.

The topological technique behind these and other related results is the following. Let $\phi$ be a convex body in $\mathbb {R}^{n}$. A field of convex bodies tangent to $\mathbb {S}^{n}$ and congruent to $\phi$ is a continuous function $\phi (u)$ defined for all $u\in \mathbb {S}^{n}$, where $\phi (u)$ is a congruent copy of $\phi$ lying in the hyperplane tangent to $\mathbb {S}^{n}$ at u. If, additionally, $\phi (u)=\phi (-u)$ for each $u\in \mathbb {S}^{n}$, we say that $\phi (u)$ is a complete turning of $\phi$ in $\mathbb {R}^{n+1}$. Clearly, a convex $(n+1)$-dimensional body with the property that all its hyperplane sections through a fixed point $p_0$ are congruent to $\phi \subset \mathbb {R}^{n}$ gives rise to a complete turning of $\phi$ in $\mathbb {R}^{n+1}$. Using the theory of fiber bundles, Mani [770] proved the following two results: (1) If $n\ge 2$ is even, the only fields of convex bodies tangent to $\mathbb {S}^{n}$ are those congruent to balls. (2) If there is a field of bodies congruent to $\phi$ tangent to $S^n$, where the group of symmetries of $\phi$ is finite, then $\mathbb {S}^{n}$ is parallelizable, that is, $n=1,3$, or 7. Furthermore, in [846] it is proved that if there is a complete turning of $\phi$ in $\mathbb {R}^{n+1}$, then $\phi$ is centrally symmetric.

Regarding the symmetries of projections and sections of convex bodies, there is a very interesting and complete survey of Ryabogin [990] in which these and many other related problems are treated. In particular, the paper of Myroshnychenko and Ryabogin [880] is very interesting; it concerns polytopes with congruent projections or sections. Finally, for more about sections and projections of convex bodies see Rogers [980] and [1077],  [1078],  [584],  [452], and [198]. For the problem of the false center of symmetry see Section 2.12.2 and the Notes of Chapter 2 regarding ellipsoids. Interesting problems and results concerning the reconstruction of convex bodies from their projections can be found in the papers of Golubyatnikov [433],  [434],  [432], and of Kuz’minykh [675]. See also Gardner’s book [401] and Section 16.4.

Exercises

• 9.1. Prove, without using Theorem 9.1.1, that if every section of a convex body $\phi$ through a point $p_0\in \mathbb {E}^n$ is a ball, then $\phi$ is a ball.

• 9.2. In the proof of Theorem 9.1.1, prove that it is enough to consider the case $n=k+1$.

• 9.3. Prove that if every section of a convex body $\phi$ is an ellipsoid, then $\phi$ is an ellipsoid.

• 9.4*. If every section of a convex body $\phi$ is a polytope, is $\phi$ a polytope?

• 9.5*. If every section of a convex body $\phi$ is centrally symmetric, is $\phi$ an ellipsoid?

• 9.6*. If every section through a fixed point of a convex body $\phi$ is centrally symmetric, is $\phi$ centrally symmetric? Is $\phi$ an ellipsoid?

• 9.7. If every hyperplane through a fixed point of a convex body $\phi$ is a hyperplane of orthogonal symmetry, is $\phi$ then a ball?

• 9.8. If every hyperplane through a fixed point of a convex body $\phi$ is a hyperplane of affine symmetry, is $\phi$ then an ellipsoid?

• 9.9*. If a 3-dimensional convex body $\phi$ has two different axes of symmetry, is $\phi$ a ball?

• 9.10*. If $\phi$ is the intersection of an infinite family of balls, is $\phi$ then spindle h-convex for some $h>0$?

• 9.11*. Is a strictly convex body the intersection of an infinite family of balls?