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

{\bf Abstract.} Let $D$ be a convex body of diameter $\delta$, where $0<\delta<\frac{\pi}{2}$, on the $d$-dimensional sphere. We prove that $D$ is of constant diameter $\delta$ if and only if it is of constant width $\delta$ in the following two cases. The first case is when $D$ is smooth. The second case is when $d=2$.


Introduction
The subject of this paper is spherical geometry (for a larger contexts see the monographs [2], [8] and [9]).
In the next section we recall the notion of a spherical convex body of constant width.
Shortly speaking, for a convex body C on the d-dimensional sphere S d and any hemisphere K supporting C we define the width of C determined by K as the thickness of any narrowest lune K ∩ K * containing C. By a body of constant width we mean a spherical convex body whose all widths are equal.
Let C ⊂ S d be a convex body of diameter δ. 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?
Our aim is to prove that every smooth spherical convex body of constant diameter δ is of constant width δ, and that a body on the two-dimensional sphere is of constant diameter δ if and only it is of constant width δ. As a consequence of these facts, the above problem remains now open only for non-smooth bodies of constant diameter below π 2 . By the way, in [3], [4] and [6] spherical bodies of constant width and constant diameter π 2 are applied for recognizing if a Wullf shape is self-dual.

On spherical geometry
The intersection of S d with any (k + 1)-dimensional Euclidean space, where 0 ≤ k ≤ d − 1, is called a k-dimensional subsphere of S d . For k = 1 we call it a great circle, and for k = 0 a pair of antipodes. If different points a, b ∈ S d are not antipodes, by the arc ab connecting them we mean this part of the great circle containing a and b, which does not contain any pair of antipodes. By the spherical distance |ab|, or shortly distance, of these points we understand the length of the arc connecting them.
By a d-dimensional spherical ball of radius ρ ∈ (0, π 2 ], or shorter a ball, we mean the set of points of S d which are at the distance at most ρ from a fixed point, called the center of this ball. For d = 2 it is called a disk, and its boundary is called a circle of radius δ. Spherical balls of radius π 2 are called hemispheres. In other words, by a hemisphere of S d we mean the common part of S d with any closed half-space of E d+1 . We denote by H(c) the hemisphere whose center is c. Two hemispheres whose centers are antipodes are called opposite hemispheres.
By a spherical (d − 1)-dimensional ball of radius ρ ∈ (0, π 2 ] we mean the set of points of a (d− 1)-dimensional great sphere of S d at the distance at most ρ from a point, called the center of this ball. The (d − 1)-dimensional balls of radius π 2 are called (d − 1)-dimensional hemispheres. Let a set C ⊂ S d does not contain any pair of antipodes. We say that C is convex if together with every two its points, C contains the whole arc connecting them. If the interior of a closed convex set C is non-empty, we call C a convex body. Its boundary is denoted by bd(C).
If a hemisphere H contains a convex body C and if p ∈ bd(H) ∩ C, we say that H supports C at p or that H is a supporting hemisphere of C at p. If exactly one hemisphere supports a convex body C at its boundary point p, we say that p is a smooth point of bd(C), and in the opposite case we say that p is an acute point of bd(C) . If every boundary point of C ⊂ S d is smooth, then C is called smooth. We call C strictly convex if bd(C) does not contain any arc. For any convex body C ⊂ S d and any hemisphere K supporting C we define the width width K (C) of C determined by K as the thickness of any narrowest lune K ∩ K * containing C (so that no lune of the form K ∩ K ′ with a smaller thickness contains C). By the thickness ∆(C) of C we mean the minimum of width K (C) over all hemispheres K supporting C. Clearly, ∆(C) is nothing else but the thickness of a "narrowest" lune containing C. We say that C is of constant width w if all its widths are w.
The above notions are given and a few properties of lunes and convex bodies in S d are presented in [5] and [7].
Lemma. Let K be a hemisphere of S d and let p ∈ bd(K). Moreover, let pq ⊂ K be an arc orthogonal to bd(K) with q in the interior of K and |pq| < π 2 . Then from amongst all the lunes of the form K ∩ M , with q in the boundary of the hemisphere M , the lune K ∩ K ⊣ such that pq is orthogonal to bd(K ⊣ ) at q has the smallest thickness.
An easy proof is left to the reader.

Spherical bodies of constant diameter
The notion of a spherical body of constant diameter is recalled in the Introduction. In this section we present a few propositions on bodies of constant diameter. Proposition 1. Every convex body D ⊂ S d of constant diameter δ < π 2 is strictly convex.
Proof. Assume the opposite that D is not strictly convex. Then bd(D) contains an arc xz.
Since p is a smooth point of bd(D), we conclude that K supports B at p.  Since our chord pp ′ of length δ is orthogonal to bd(K) at p and to bd(K ′ ) at p ′ , by the definition of the thickness of a lune we see that the lune K ∩ K ′ has thickness |pp ′ | = δ.
Consequently, by D ⊂ K ∩ K ′ we conclude that width K (D) ≤ δ.
By Lemma the lune K ∩ K ′ is the narrowest lune from the family L 1 of lunes of the form the narrowest lune from the family L 2 of all lunes the form K ∩ M containing D. Since every lune from L 2 contains a lune from L 1 , every lune from L 2 is of thickness at least δ. Consequently, From the above two paragraphs we conclude that width K (D) = δ.

Two cases in which a spherical convex body of constant diameter is of constant width
Since the question is answered for δ ≥ π 2 and since, as mentioned in the Introduction, every spherical body of constant width is of constant diameter, now we concentrate on checking when a spherical body of constant diameter δ < π 2 is of constant width. The following theorem gives a partial answer. It results immediately from Proposition 3 and from the fact that every body of constant width δ is of constant diameter δ.
A spherical smooth convex body on S d is of constant diameter δ if and only if it is of constant width δ.
Below is our main theorem. Since in its proof we apply polar sets, let us recall this notion.
For a convex body C ⊂ S d by its polar we mean the set C • = {r : C ⊂ H(r)}. It is easy to show that C • is a convex body. Recall that bd(C • ) is the set of points r such that H(r) is a supporting hemisphere of C.
Theorem 2. Let 0 < δ < π 2 . A convex body on the two-dimensional sphere is of constant diameter δ if and only it is of constant width δ.
Proof. In [7] there is observed that every body of constant width on S d is a body of constant diameter.
It remains to show that every body D ⊂ S 2 of constant diameter δ < π 2 is of constant width δ, i.e., that width K (D) = δ for every supporting hemisphere K of D.
By Proposition 1 there is exactly one point p of support of D by K.   What is more, s ∈ P . The reason is that if s ∈ F \ P , then the disk of center s and radius δ contains D. Consequently, the hemisphere supporting the ball bounded by F at p is a supporting hemisphere of D at p. This contradicts the fact that H 1 and H 2 are the first and the last supporting hemispheres of D at p. A forthcoming paper is devoted to show the analogous facts as in Propositions 1-3 and Theorems 1-2 in E d (they are true also in the hyperbolic space). The proofs of these theorems apply the parallelism, which cannot be used here for spherical bodies.