Spindle Convexity

Abstract

Let h be a positive real number and let p and q be two points in Euclidean n-space $\mathbb {E}^n$, no more than 2h apart. The h-interval determined by this pair of points is the intersection of all balls of radius h that contain p and q. We say that a set $\phi$, with diameter less than or equal to 2h, is spindle h-convex if given a pair of points p and q in $\phi$, the h-interval they determine is also in $\phi$.

There is geometry in the humming of the strings, there is music in the spacing of the spheres.

Pythagoras

Appendices

Notes

Ball Convexity and Spindle Convexity

Replacing intersections of half-spaces by intersections of balls, one gets possibilities to extend the classical convexity notion in a natural way to generalized convexity notions. It is the spherical intersection property of constant width sets that motivates also the study of notions like ball convexity, spindle convexity, and ball polytopes. A subset S of ${\mathbb E^n}$ is called ball convex if it coincides with its ball hull, which is the intersection of all balls of fixed radius containing S. On the other hand, S is said to be spindle convex if, for all $x, y \in S$, S covers the whole spindle of x and y, i.e., the intersection of all balls of fixed radius containing x, y. Spindle convex sets are not necessarily closed, but ball convex sets are. In ${\mathbb E^n}$, closed sets are spindle convex iff they are ball convex, see Exercise 10.16.

Since these notions are also important for Banach space theory, we remark that this coincidence does not hold in normed spaces (see Corollary 3.4 of [118], Corollaries 3.13 and 3.15 of [684], and also [579] for situations that are even more general). To avoid confusion by homonymous usage of names, we shortly mention another concept which is not discussed in this book. Namely, a set is (also) called ball convex if with any finite number of points it contains the intersection of all balls of arbitrary radii containing the points; the ball convex hull of a set S is then the intersection of all ball convex sets containing S (see e.g., [688]). It is obvious that in the Euclidean case this type of ball convexity coincides with linear (i.e., usual) convexity. With the name “Überkonvexität” (overconvexity), the notion of spindle convexity was introduced by Mayer [804], and different older contributions (partially also referring to notions like ball convexity in the sense of ball hulls) are [192],  [1150],  [265],  [921],  [126],  [128],  [1017], and [124]. One starting point with such notions in discrete geometry was given by Fejes Tóth [346]. All these concepts naturally fit into the framework of generalized convexity notions as they are discussed in § 9 of the famous survey [278]. We discuss now some more recent results on spindle convex sets. First, we note that different names like K-convexity, r-convexity, R-convexity, hyperconvexity, overconvexity (and perhaps more) are also used in the literature. In [114], results of Schramm on the illumination number of bodies of constant width (=smallest number of smaller homothets needed to cover such a body) are extended to spindle convex sets. The paper [371] contains analogues of important inequalities (like the Blaschke–Santaló inequality and Ball’s reverse isoperimetric inequality) for spindle convexity. Dowker’s theorem from the theory of packings and coverings says that the maximum area of m-gons inscribed in a convex disk K is a concave function of m, while the minimum area of m-gons circumscribed about K is a convex function of m. In [340], some Dowker-type theorems for spindle convex disks are proved. In [96], results about the Gaussian curvature of spindle convex sets are obtained, and we also refer to [122], where the usual notion of starshapedness is extended to spindle starshapedness via a spindle convex variant of visibility. The authors then establish respective analogues of the well-known theorems by Krasnosel’skiĭ, Carathéodory, and Klee, as well as spindle convex variants of some more recent results, including also a version of the art gallery problem from computational geometry. The papers [118] and [670] are basic for the study of spindle convexity in connection with the related notion of ball polytopes. Further interesting results on spindly convexity were obtained in [112] and [114], see also [916]. Also Chapters 7 and 8 of the monograph [115] refer to spindle convexity and ball polytopes.

Ball Polytopes

A ball polytope is a nontrivial intersection of finitely many unit balls (or balls of equal radii) in ${\mathbb E}^n$. There are many applications and relations between the notions of spindle convexity, ball convexity, ball polytopes, ball hulls, and the spherical intersection property, see again the two papers [118] and [671]. For example, in [118] analogues of the classical theorems of Kirchberger, Carathéodory, and Steinitz for spindle convex sets are proved, and the face structure of ball polytopes is studied. This is continued in [671] in a very detailed way. Since diagonal faces occur and two faces may intersect in more than one edge, the respective graph of 3-dimensional ball polytopes is 2-, but not necessarily 3-connected. The authors of [671] derive also results on characterizations of finite sets V of diameter 1, say, in which diameters (now meant as segments of length 1 in a geometric graph with vertices from V) occur a maximal number of times. This is related to topics like the Vázsonyi problem and the Grünbaum–Heppes–Straszewicz theorem, see Section 6.3. In addition, the results of Sallee [998] on Reuleaux polytopes are discussed again in [671], in the light of the framework of ball polytopes. Further properties of ball polytopes refer to rigidity (see [121]) and to the Kneser–Poulsen conjecture on volumes of unions and intersections of finite systems of balls (see, for example, [112]). There are more papers on ball polytopes and the above- generalized convexity notions (most of them cited in the references listed here), but since their contents are too distant from the concept of constant width, we stop here.

Circular Intersection Properties

The planar version of the spherical intersection property is usually called circular intersection property. Closely related is the so-called weak circular intersection property introduced in [668]. A set S of diameter 1 has this property, if the intersection of all unit circles, whose centers belong to S, is of constant width. A finite example is the vertex set of a Reuleaux polygon. The authors of [668] determine the smallest number of points necessary to complete a finite planar set of given diameter such that the resulting set (obtained as the described intersection) has the weak circular intersection property. This number is given in terms of the diameter graph of the set. In [210], the fact is recalled that, given a set of points, the Minkowski sum of the intersection of all disks of fixed radius centered at these points and the intersection of all disks of this radius, which contain the points, is a constant width set. This is used to prove a nice conjecture comparing the two areas of intersections of two families of equal disks. For the spherical intersection property in higher dimensions and its numerous relations to other notions (like completeness, constructions, etc.) discussed in this book, we refer to the Sections 4.5, 7.6, and 10.7, where also related references are collected.

Figure 6.8

Exercises

• 6.1. Prove that h-intervals are spindle h-convex sets and that every spindle h-convex body is strictly convex.

• 6.2. Prove that the h-interval determined by the points p and q is the union of all short circular arcs of radius $\rho$, $h\le \rho \le \infty$, through p and q.

• 6.3. Prove that for $h^\prime \ge h$ spindle h-convex sets are $h^\prime$-convex.

• 6.4. Prove that the intersection of a family of spindle h-convex sets is spindle h-convex.

• 6.5. Prove that balls of radius h are the only spindle h-convex sets of diameter h.

• 6.6. Let $\Phi$ be a spindle h-convex body and suppose $p\notin \Phi$. Prove that there is a ball B(h) of radius h with the property that $\Phi \subset B(h)$, but $p\notin B(h)$.

• 6.7. Let $\Phi$ be a convex body. Prove that $\bigcap _{p\in \Phi } B(p,h)=\bigcap _{p\in {{\,\mathrm{\mathrm {bd}}\,}}\Phi } B(p, h)$.

• 6.8. Prove that $\Phi \subset \bigcap _{p\in {{\,\mathrm{\mathrm {bd}}\,}}\Phi } B(p, h)$ if and only if the diameter of $\Phi$ is smaller than or equal to h.

• 6.9. Prove that the property $\bigcap _{p\in {{\,\mathrm{\mathrm {bd}}\,}}\Phi } B(p, h) \subset \Phi$ is equivalent to the spherical intersection property; that is, if a ball B(p, h) contains $\Phi$, then $p\in \Phi$.

• 6.10*. Let $L_a$ and $L_b$ be two parallel lines at distance h apart and consider points $a\in L_a, b\in L_b$ such that the segment ab is orthogonal to these two lines. Consider a curve $\mathcal{C}$ between $L_a$ and $L_b$, to the right of ab tangent to $L_a$ at a, tangent to $L_b$ at b (see Figure 6.8) and such that through every point $c\in \mathcal{C}$ there is a support h-circle to $\alpha$ of c. Complete the curve $\alpha$ to a closed curve $\tilde{\alpha }$ by adding the centers of all these support circles. Is the curve $\tilde{\alpha }$ a curve of constant width h?

• 6.11. Prove that a convex body of constant width h is the intersection of countably many balls of radius h.

• 6.12. Let $S\subset \mathbb {E}^n$ be a set of diameter smaller than h. Then the diameter of ${{\,\mathrm{\mathrm {cc}}\,}}_h(S)$ is equal to the diameter of S.

• 6.13. Prove the Vázsonyi problem in the plane. That is, a finite set V of cardinality m and diameter h has at most m diameters, and the diameter is attained m times if and only if $V\subset {{\,\mathrm{\mathrm {bd}}\,}}{\mathcal {B}}(V, h)$ and every singular point of the boundary of ${\mathcal {B}}(V, h)$ belongs to V.

• 6.14. Prove that a 2-dimensional Reuleaux polytope is a Reuleaux polygon.

• 6.15. Prove that if $\mathcal{J}=pq$ is a dual binormal, where $p\in \widetilde{ab}$ and q is in its dual edge $\widetilde{xy}$, then the length of $\mathcal{J}$ is equal to $2\sqrt{h^2-\frac{t^2}{2}}-\sqrt{h^2-\frac{t^2}{4}}$, where $|x-a|=|x-b|=|y-a|=|y-b|=h$.

• 6.16. Let $n\ge 2$ be an integer and $0<\alpha < 2$. Construct the graph $G(n,\alpha )$ on the points of $\mathbb {S}^{n-1}$ by connecting two of them if and only if their distance is at least $\alpha$. The graph $G(n,\alpha )$ obtained this way has chromatic number at least $n+1$. Prove that this fact is equivalent to the Theorem of Borsuk [168], see Section 15.3.1.