Basic Properties of Bodies of Constant Width

Abstract

One of the essential characteristics of bodies of constant width is that, like balls, they have a diameter in every direction. Diameters are those chords of a convex body that have maximum length, and it is their behavior which gives constant width bodies their basic properties. Unlike the diameters of a ball, those of a body of constant width do not always meet at a single point, but when they do so, it is because the body is indeed a ball.

The only royal road to elementary geometry is ingenuity.

Eric Temple Bell

Appendices

Notes

Based on the obvious relation between width function and support function of a convex body, it is true that (even in any Minkowski space) a set $\Phi$ is of constant width $h > 0$ if and only if its central symmetral $\frac{1}{2}(\Phi +(-\Phi ))$ coincides with $\frac{h}{2} B^n(o, 1)$; see, e.g., [1113]. In [310], Eggleston proved that $\Phi$ is of constant width if and only if for each pair $H_1,H_2$ of parallel support hyperplanes there is a diametral chord of $\Phi$ generated by, and orthogonal to $H_1,H_2$, from which also other parts of Theorem 3.1.4 follow. Thus, it is necessary and sufficient that all diametral chords have length h, i.e., that $\Phi$ is of constant diameter. Regarding this formulation, the planar case was solved in [1204], p. 73, and Burke [197] used this for constructions of constant width curves. In [511], Hammer and Sobczyk mentioned this fact explicitly when studying outwardly simple line families built from prolonged diametrical chords (however, Hammer [502] observed that this characterization fails in 3-space). Theorem 3.1.3, that ensures that every constant width body has the binormal property, was mentioned in [308], p. 126; Besicovitch [104] confirmed the planar case of the converse, and one can easily see the higher dimensional analogue by the invariance of that property when projecting such sets orthogonally into 2-dimensional planes. Eggleston [310] proved that any two parallel normals coincide if and only if a convex body has constant width. For all these comments see also Sections 3.1 and 3.5 for Euclidean space and Sections 10.1 and 10.2 for Minkowski spaces.

In [663] the following was derived: For each set Q of n lines through the origin o in ${\mathbb E}^n$, there exists a convex body K such that $o \in \mathrm{int} K$, and the double normals of K are exactly the intersections $K \cap Q$. For $n \le 3$, the set of all lengths of double normals is of measure zero, but for $n \ge 4$ there exists an n-dimensional convex body K, not of constant width, such that every width of K is attained as the length of some double normal, and such that the set of all directions of double normals of K contains an arc connecting a minimum width direction to a maximum width direction. A further nonintuitive property of normals was shown by Heil (see [525] and [526]): Every 3-dimensional constant width body contains either a point through which infinitely many normals pass, or an open set of points through each of which at least 10 normals pass. By the double-normal property of diametral chords a light ray reflecting internally at the boundary of a smooth constant width body will cross a diametral chord repeatedly if any segment of its way lies in that chord. Generalizing a result from [1067] obtained for the 3-dimensional situation, in [1068] it was shown that an n-dimensional smooth convex body is a ball iff each path of a light ray reflecting inside that body lies in a 2-flat. Of course, this is related to billiard problems (see also Notes in chapter 5 and 17.2). It is also proved in [1068] that the successive reflection points inside a smooth constant width curve in the plane and not on a double normal follow a clockwise or counterclockwise order, that this property even characterizes such curves, and that therefore the path of such light rays is never ergodic. Somehow related to double normals is the usage of the farthest point mapping. In [572] the mapping F, associating to each point x of a convex hypersurface the set of all points at maximal intrinsic distance from x, is used to provide two large classes of hypersurfaces with the mapping F single-valued and involutive, and to show that if the mapping F of the double of a convex body satisfies the above properties, then this body has to be a smooth constant width set; a partial converse of this result is verified, too. The following results refer to potential theory see Chapter 18, but we mention them also here since they also use the farthest point mapping in the above sense. Let E be a compact planar set containing more than one point. The function $d_E(z) = \mathbf{max}_{t \in E} \Vert z-t\Vert$ is strictly positive, and log$d_E$ is clearly subharmonic and the logarithmic potential of a unique probability measure $\sigma _E$ with unbounded support. Various properties of $d_E$ and $\sigma _E$ are established in [705],  [396], and [397], e.g., that $\sigma _E(E)=\frac{1}{2}$ holds for sets E of constant width and some converse. Alternative proofs and generalizations are presented in [611], see also [609]. At the end of [611] also 3-dimensional examples for E are given, among them constant width sets, such as rotated Reuleaux triangles and Meissner bodies, see also [609].

In [94] it was proved that a planar convex body is of constant width if and only if it is strictly convex and has the property of constant minimum width, see Section 3.2 and Theorem 7.4.3. Furthermore, in [866] it was proved that a convex set in the plane is of constant width iff its boundary is the image of a continuous map of a circle, such that diameters are mapped to double normals.

Concerning the perimeter of the shadow boundary (see Section 2.12.2) of a 3-dimensional body of constant width 1, Makeev [761] was able to prove that it is always smaller or equal that $\sqrt{2}\pi$.

The following paragraph refers to projections of constant width bodies and is therefore strongly related to Chapter 13 and its notes. Already in [160] p. 127, one can find the well-known theorem that if for some k with $n> k > 1$ and $n \ge 3$ all orthogonal projections of an n-dimensional convex body K onto k-dimensional subspaces are of constant width, then K is of constant width, too. One can take also other properties of projections into consideration. For example, Minkowski proved that a 3-dimensional convex body K is of constant width if and only if for all directions u its orthogonal projections $K_u^\perp$ in planes have equal perimeter (see [888],  [160], p. 136, and [357], p. 39), i.e., if and only if that body K is of constant girth (to take another name for this). For a presentation of this theorem and its proof in modern terms we refer to [464], p. 219–221. See also Section 13.1. Denoting by $M_{n-1}(K_u^\perp )$ the mean width of the orthogonal $(n-1)$-projection $K_u^\perp$ of K, the following theorem generalizes Minkowski’s theorem (note that $M_{n-1}(K_u^\perp )$ playing the role that the perimeter has in Minkowski’s theorem for $n = 3$): If K and L are convex bodies in ${\mathbb E}^n, n > 2,$ such that for all directions u the equality $M_{n-1}(K_u^\perp ) = M_{n-1}(L_u)$ holds, then K and L have equal width functions; if, in particular, $M_{n-1}(K_u^\perp )$ is constant, then K is of constant width ([464], p. 221). A stability version of Minkowski’s generalized theorem is given in [464], p. 225. We come back now to three dimensions. If we multiply the perimeter of the projection with the width of K in the same direction u, we get the “lateral surface area” of a compact cylinder circumscribed about K and with generators which are parallel to u. Firey [354] showed that K has constant width iff all such cylinders circumscribed about K have equal lateral surface area. See Section 13.1.3. Weissbach [1189] proved that for $n \ge 4$ there is a body K of constant width 1 in ${\mathbb E}^n$ such that the circumradius of the orthogonal projection $K_u$ of K in direction u is strictly greater than $\frac{1}{2}$ for all directions u. In a separate way, he confirmed this also for $n=3$, reproving therefore a crucial lemma used in [309] for constructing minimal universal covers with arbitrarily large diameter. Continuing [309] and [1189], Brandenberg and Larman [181] proved that for any $n \ge 3$ there exists an n-dimensional body of constant width such that any of its 2-dimensional projections is not spherical. Calling such a constant width body totally nonspherical, they showed that the circumradius of each 2-dimensional projection of any totally nonspherical constant width body is larger than half the diameter of it. For generalizing the concept of constant width suitably, the following notions are used in [176]: the outer j-radius, $j=1,\dots , n-1$, of a convex body $K \subset {\mathbb E}^n$ is the minimum of the circumradii of the projections of K onto all j-dimensional subspaces, and the inner j-radius of K is the radius of the largest j-dimensional ball that fits into K. The authors of [176] nicely construct in any dimension $n \ge 2$ nonspherical convex bodies of constant inner and outer j-radii. For a given convex body K in ${\mathbb E}^n$, Kiderlen [624] considered the Minkowski average (i.e., the convex body $P_k(K)$ whose support function is defined by integration of support functions of the orthogonal projections of K onto k-dimensional subspaces with respect to the rotational invariant probability measure on the corresponding Grassmannian manifold) of all such k-projections of K. Among other results, he confirmed that $P_{2k+1}(K)$ determines K among all bodies of constant width. Let K and L be convex bodies in ${\mathbb E}^n$, symmetric with respect to the origin, and suppose that $\Vert M(K_u^\perp )- M (L_u^\perp )\Vert _2 \le \varepsilon$ for some $\varepsilon > 0$. Here M again denotes the mean width, and $\Vert \cdot \Vert _2$ is the $L_2$-norm for functions on the unit sphere. It is proved in [441] that the $L_2$-distance $h_2(K, L)$ of K and L can be estimated by $h_2(K, L) \le \frac{1}{2} \varepsilon ^{\frac{2}{n}}(\lambda _n(\delta (K)+\delta (L))+\varepsilon ^2)^{(n-2)/2n}$, where $\lambda _n$ is an explicitly given constant and $\delta =\kappa _d^{-1} W_{n-1}^2 - W_{n-2}$ ($\kappa _n$ denotes the volume of the unit ball, and $W_k$ is written for the k-th quermassintegral). It follows that if the mean width $M(K_u^\perp )$ of $K_u^\perp$ is nearly constant, then there is a constant width set near K in the Hausdorff metric. For results on recovering the shape of at least 3-dimensional convex bodies from their projections onto 2-planes, Golubyatnikov [435] had to exclude that all these projections are of constant width. Continuing these investigations in the spirit of geometric tomography, Groemer [462] proved that a convex body K is of constant width iff all its orthogonal projections onto hyperplanes of a hyperplane bundle (i.e., of a family of hyperplanes having a line in common such that each point of ${\mathbb E}^n$ belongs to some of them, where the position of this line is arbitrary) are of constant width. For $n = 3$ and strictly convex bodies this uniqueness is actually valid if the definition of a hyperplane bundle the condition of having a common line is suitably weakened (see [848], where also further related results are obtained). Besides such uniqueness results, Groemer [462] also derived analogous stability results. In connection with polyhedral universal covers, Makeev [757] investigated aspherical orthogonal projections of constant width bodies. In [762], the same author studied shadow boundaries of constant width bodies in ${\mathbb E}^3$. He showed that they form rectifiable spatial curves of length at most $\sqrt{2} \pi$ which cannot be improved. And there is one direction such that the corresponding orthogonal projection of such a body has length at most $\sin (\pi /10)+\sin (\pi /20)$. Weakly related is the paper [826], where support functions of constant width bodies and projections of support functions of convex sets play a role. We continue with in- and circumspheres of bodies of constant width. It is well known that the insphere and the circumsphere of a body of constant width h are concentric, and that their respective radii r and R satisfy $r + R = h$ as well as $h(1-(\frac{n}{2}(n+1))^{\frac{1}{2}}) \le r$ and $R \le h(\frac{n}{2}(n+1))^{\frac{1}{2}}$, see [840],  [308], and [821]. The extension to Minkowski spaces was given in [231]. Scott [1052] proved that a convex set in ${\mathbb E}^n$ is contained in a set of constant width with the same diameter and circumradius, and he derived from this several related inequalities, see Section 14.4. And the following nice theorem was proved in [1159]: If C is a set of diameter h contained in a ball B in ${\mathbb E}^n$, then there exists a set $\Phi$ of constant width h such that $C \subset \Phi \subset B$. More on inspheres and circumspheres and the respective radii we discuss in the Notes of Chapter 14.

Exercises

• 3.1. Prove that a body has constant width if and only if it has a diameter in every direction.

• 3.2. Prove that if $\Phi$ and $\Psi$ are bodies of the same constant width and $\Psi \subset \Phi$, then $\Psi =\Phi$.

• 3.3. Prove that a body of constant width h is the intersection of a family of solid balls of radius h.

• 3.4. Prove that in a convex body every diameter is a binormal.

• 3.5. Prove that in a body of constant width every binormal is a diameter.

• 3.6. Let $\Phi$ be a body of constant width h, and p be a vertex point in boundary of $\Phi$. Prove that there is an open subset U of the sphere with center p and radius h such that $U\subset$ bd $\Phi$.

• 3.7*. Let $\phi \subset \mathbb {E}^n$ be a convex body with the following property: there exists a point $o\in$ int$\phi$ such that for every $m\in {{\,\mathrm{\mathrm {bd}}\,}}\phi$ the hyperplane through m, orthogonal to om, is a support hyperplane of $\phi$. Prove that $\phi$ must be a ball. Note that the proof is easier if we assume differentiability.

• 3.8. Suppose $\phi$ is a convex body with the property that given two parallel support hyperplanes H and $H^\prime$, we can choose $p\in H$ and $q\in H^\prime$ in such a way that pq is orthogonal to H and $H^\prime$. Prove that $\phi$ has constant width.

• 3.9. Prove that a convex body $\Phi$ has constant width if and only if any two parallel normals coincide.

• 3.10. Prove that a 3-dimensional convex body $\Phi$ is of constant width if and only if the line perpendicular to each pair of nonparallel normals intersects $\Phi$.

• 3.11. Let $\phi \subset \mathbb {E}^n$ be a convex body, and let $u\in \mathbb {S}^{n-1}$ be a direction. Consider the chord pq of $\phi$ with the maximal length between all chords of $\phi$ in the direction u. Prove that pq is a diametral chord. Note that this implies that there is a diametral chord parallel to every direction.

• 3.12*. Prove that if $\phi$ is strictly convex, then the inverse Gauss map $\gamma :\mathbb {S}^{n-1}\rightarrow$ bd$\phi$ is a continuous map.

• 3.13. Prove that if $\phi$ is strictly convex and $\gamma :\mathbb {S}^{n-1}\rightarrow {{\,\mathrm{\mathrm {bd}}\,}}\phi$ is the inverse Gauss map, then $\gamma ^{-1}(p)$ is spherically convex, for every boundary point p of $\Phi$.

• 3.14. If $\phi$ is strictly convex, $\gamma :\mathbb {S}^{n-1}\rightarrow$ bd$\phi$ is the inverse Gauss map and p is a vertex point of $\Phi$, then $\gamma ^{-1}(p)$ has interior points. Use this to prove that a strictly convex body has countably many vertex points.

• 3.15. Prove that a continuous function $f:\mathbb {S}^{n-1}\rightarrow \mathbb {E}$ is a constant function if and only if the restriction of f to every great circle of $\mathbb {S}^{n-1}$ is a constant function.

• 3.16. Give an example of a convex body in which the minimum width w(p) is not a continuous function of p.

• 3.17. Prove that the n-dimensional cube and the common part of the unit solid n-sphere with the first closed octant have constant minimum width.

• 3.18*. Give an example of a regular point p in the boundary of a convex body with constant minimum width whose normal is not a thickness binormal.

• 3.19. Prove that if every regular point p in the boundary of a convex body $\Phi$ is a thickness binormal, then $\Phi$ has constant minimum width.

• 3.20. Prove that a square is tight. Prove that any regular polygon is tight.

• 3.21. Give an example of a figure of constant minimum width which is not tight.

• 3.22. Consider $X\subset \mathbb {E}^n$ to be a compact set, not necessarily convex, of constant width. Prove that every support hyperplane intersects X in a single point. Use this to prove that the boundary of the convex hull of X is contained in X.

• 3.23. Let $X\subset \mathbb {E}^n$ be a set of constant width and, suppose that $\mathbb {E}^n \backslash X$ has only one component. Prove that X is convex.