Spherical bodies of constant width

Abstract

The intersection L of two different non-opposite hemispheres G and H of the ddimensional unit sphere S d is called a lune. By the thickness of L we mean the distance of the centers of the (d − 1)-dimensional hemispheres bounding L. For a hemisphere G supporting a convex body C ⊂ S d we define width G (C) as the thickness of the narrowest lune or lunes of the form G ∩ H containing C. If width G (C) = w for every hemisphere G supporting C, we say that C is a body of constant width w. We present properties of these bodies. In particular, we prove that the diameter of any spherical body C of constant width w on S d is w, and that if w < π 2 , then C is strictly convex. Moreover, we check when spherical bodies of constant width and constant diameter coincide. Mathematics Subject Classification. 52A55.A spherical ball B ρ (x) of radius ρ ∈ (0, π 2 ], or a ball for short, is the set of points of S d at distances at most ρ from a fixed point x, which is called the center of this ball. An open ball (a sphere) is the set of points of S d having distance smaller than (respectively, exactly) ρ from a fixed point. A spherical ball of radius π 2 is called a hemisphere. So it is the common part of S d and a closed half-space of E d+1 . We denote by H(m) the hemisphere with center m. Two hemispheres with centers at a pair of antipodes are called opposite.A spherical (d − 1)-dimensional ball of radius ρ ∈ (0, π 2 ] is the set of points of a (d − 1)-dimensional great sphere of S d which are at distances at most ρ from a fixed point. We call it the center of this ball. The (d − 1)-dimensional balls of radius π 2 are called (d − 1)-dimensional hemispheres, and semicircles for d = 2.A set C ⊂ S d is said to be convex if no pair of antipodes belongs to C and if for every a, b ∈ C we have ab ⊂ C. A closed convex set on S d with nonempty interior is called a convex body. Some basic references on convex bodies and their properties are [4], [9] and [10]. A short survey of other definitions of convexity on S d is given in Section 9.1 of [2].Since the intersection of every family of convex sets is also convex, for every set A ⊂ S d contained in an open hemisphere of S d there is the smallest convex set conv(A) containing Q. We call it the convex hull of A.Let C ⊂ S d be a convex body. Let Q ⊂ S d be a convex body or a hemisphere. We say that C touches Q from inside if C ⊂ Q and bd(C)∩bd(Q) = ∅. We say that C touches Q from outside if C ∩ Q = ∅ and int(C) ∩ int(Q) = ∅. In both cases, points of bd(C) ∩ bd(Q) are called points of touching. In the first case, if Q is a hemisphere, we also say that Q supports C, or supports C at t, provided t is a point of touching. If at every boundary point of C exactly one hemisphere supports C, we say that C is smooth. We call e ∈ C an extreme point of C if C \ {e} is convex.If hemispheres G and H of S d are different and not opposite, then L = G∩H is called a lune of S d . This notion is considered in many books and papers (for instance, see [12]).

Reduced spherical convex bodies

“…Assume that ∆(R) ≥ π 2 . By Theorem 4.3 of [12], the body R is of constant width ∆(R). There is an arc pq ⊂ R of length equal to diam(R).…”

“…Clearly, p ∈ bd(R). Take a lune L from Theorem 5.2 of [12] such that p is the center of a semicircle bounding L. Denote by s the center of the other semicircle S bounding L. By the third part of Lemma 3 in [8], we have |px| < |ps| for every x ∈ S different from s. Hence |px| ≤ |ps| for every x ∈ S. So also |pz| ≤ |ps| for every z ∈ L. In particular, |pq| ≤ |ps|. Since diam(R) = |pq| and ∆(R) = ∆(L) = |ps|, we obtain diam(R) ≤ ∆(R).…”

When is a spherical body of constant diameter of constant width?

“…Recall that bd(C • ) is the set of points r such that H(r) is a supporting hemisphere of C. Proof. In [7] there is observed that every body of constant width on S d is a body of constant diameter.…”

“…If the spherical distance |pq| of points p, q ∈ C is δ, we call pq a diametral chord of C and we say that p, q are diametrically opposed points of C. Clearly, p, q ∈ bd(C). After Part 4 of [7] we say that a convex body D ⊂ S d of diameter δ is of constant diameter δ provided for every point p ∈ bd(D) there exists at least one point p ′ ∈ bd(D) such that |pp ′ | = δ (in other words, that pp ′ is a diametral chord of D). For the known analogous notion in E d see [1].…”

“…Recall that in [7] it is proved that a convex body on S d is of diameter δ ≥ π 2 is of constant diameter if and only if it is a body of constant width δ. Moreover, there is observed that the "if" part holds also for δ < π 2 , and the problem is put if every spherical body of constant diameter δ < π 2 on S d is a body of constant width δ?…”