aa r X i v : . [ m a t h . M G ] A p r COMPLETE SPHERICAL CONVEX BODIES
MAREK LASSAK
Abstract.
Similarly to the classic notion in Euclidean space, we call a set on the sphere S d complete, if adding any extra point increases the diameter. Complete sets are convexbodies of S d . We prove that on S d complete bodies and bodies of constant width coincide. On spherical geometry
Let S d be the unit sphere in the ( d + 1)-dimensional Euclidean space E d +1 , where d ≥
2. By a great circle of S d we mean the intersection of S d with any two-dimensionalsubspace of E d +1 . The common part of the sphere S d with any hyper-subspace of E d +1 iscalled a ( d − -dimensional great sphere of S d . By a pair of antipodes of S d we mean anypair of points of intersection of S d with a straight line through the origin of E d +1 .Clearly, if two different points a, b ∈ S d are not antipodes, there is exactly one greatcircle containing them. By the arc ab connecting a and b we mean the “smaller” part ofthe great circle containing a and b . By the spherical distance | ab | , or shortly distance , ofthese points we understand the length of the arc connecting them. The diameter diam( A )of a set A ⊂ S d is the number sup a,b ∈ A | ab | . By a spherical ball B ρ ( r ) of radius ρ ∈ (0 , π ],or shorter a ball , we mean the set of points of S d having distance at most ρ from a fixedpoint, called the center of this ball. Spherical balls of radius π are called hemispheres .Two hemispheres whose centers are antipodes are called opposite hemispheres .We say that a subset of S d is convex if it does not contain any pair of antipodes andif together with every two points a, b it contains the arc ab . By a convex body , or shortly body , on S d we mean any closed convex set with non-empty interior.Recall a few notions from [6]. If for a hemisphere H containing a convex body C ⊂ S d we have bd( H ) ∩ C = ∅ , then we say that H supports C . If hemispheres G and H of S d aredifferent and not opposite, then L = G ∩ H is called a lune of S d . The ( d − L and contained in G and H , respectively, are denoted by G/H and
H/G . We define the thickness of a lune L = G ∩ H as the spherical distanceof the centers of G/H and
H/G . For a hemisphere H supporting a convex body C ⊂ S d we define the width width H (C) of C determined by H as the minimum thickness of a luneof the form H ∩ H ′ , where H ′ is a hemisphere, containing C . If for all hemispheres H supporting C we have width H (C) = w, we say that C is of constant width w . MAREK LASSAK Spherical complete bodies
Similarly to the traditional notion of a complete set in the Euclidean space E d (forinstance, see [1], [2], [3] and [10]) we say that a set K ⊂ S d of diameter δ ∈ (0 , π ) is complete provided diam( K ∪ { x } ) > δ for every x K. Theorem 1.
Arbitrary set of a diameter δ ∈ (0 , π ) on the sphere S d is a subset of acomplete set of diameter δ on S d . We omit the proof since it is similar to the proof by Lebesque [9] in E d (it is recalled inPart 64 of [1]). Let us add that earlier P´al [12] proved this for E by a different method.The following fact permits to use the term a complete convex body for a complete set. Lemma 1.
Let K ⊂ S d be a complete set of diameter δ . Then K coincides with theintersection of all balls of radius δ centered at points of K . Moreover, K is a convex body.Proof. Denote by I the intersection of all balls of radius δ with centers in K .Since diam( K ) = δ , then K is contained in every ball of radius δ whose center is at apoint of K . Consequently, K ⊂ I .Let us show that I ⊂ K , so let us show that x K implies x I . Really, from x K we get | xy | > δ for a point y ∈ K , which means that x is not in the ball of radius δ andcenter y , and thus x I .As an intersection of balls, K is a convex body. (cid:3) Lemma 2. If K ⊂ S d is a complete body of diameter δ , then for every p ∈ bd( K ) thereexists p ′ ∈ K such that | pp ′ | = δ .Proof. Suppose the contrary, i.e., that | pq | < δ for a point p ∈ bd( K ) and for every point q ∈ K . Since K is compact, there is an ε > | pq | ≤ δ − ε for every q ∈ K .Hence, in a positive distance below ε from p there is a point s K such that | sq | ≤ δ for every q ∈ K . Thus diam( K ∪ { s } ) = δ , which contradicts the assumption that K iscomplete. Consequently, the thesis of our lemma holds true. (cid:3) For different points a, b ∈ S d at a distance δ < π from a point c ∈ S d define the pieceof circle P c ( a, b ) as the set of points v ∈ S d such that cv has length δ and intersects ab .We show the next lemma for S d despite we apply it later only for S . Lemma 3.
Let K ⊂ S d be a complete convex body of diameter δ . Take P c ( a, b ) with | ac | and | bc | equal to δ such that a, b ∈ K and c ∈ S d . Then P c ( a, b ) ⊂ K .Proof. First let us show the thesis for a ball B of radius δ in place of K . There is unique S ⊂ S d with a, b, c ∈ S . Consider the disk D = B ∩ S . Take the great circle containing P c ( a, b ) and points a ∗ , b ∗ of its intersection with the circle bounding D . There is unique c ∗ ∈ S such that P c ( a, b ) ⊂ P c ∗ ( a ∗ , b ∗ ). Clearly, P c ∗ ( a ∗ , b ∗ ) ⊂ D ⊂ B . Hence P c ( a, b ) ⊂ B. By the preceding paragraph and Lemma 1 we obtain the thesis of the present lemma. (cid:3)
OMPLETE SPHERICAL CONVEX BODIES 3 Complete and constant width bodies on S d coincide Here is our main result presenting the spherical version of the classic theorem in E d proved by Meissner [11] for d = 2 , d . Theorem 2.
A convex body of diameter δ on S d is complete if and only if it is of constantwidth δ .Proof. ( ⇒ ) Prove that if K ⊂ S d of diameter δ is complete, then K is of constant width δ .Suppose the opposite, i.e., that width I (K) = δ for a hemisphere I supporting K . ByTheorem 3 and Proposition 1 of [6] width I (K) ≤ δ . So ∆( K ) < δ . By lines 1-2 of p. 562of [6] the thickness of K is equal to the minimum thickness of a lune containing K . Takesuch a lune L = G ∩ H , where G, H are different and non-opposite hemispheres. Denoteby g, h the centers of
G/H and
H/G , respectively. Of course, | gh | < δ . By Claim 2 of [6]we have g, h ∈ K . By Lemma 2 there exists a point g ′ ∈ K in the distance δ from g . Sincethe triangle ghg ′ is non-degenerate, there is a unique two-dimensional sphere S ⊂ S d containing g, h, g ′ . Clearly, ghg ′ is a subset of M = K ∩ S . Hence M is a convex bodyon S . Denote by F this hemisphere of S such that hg ′ ⊂ bd( F ) and g ∈ F . There is aunique c ∈ F such that | ch | = δ = | cg ′ | . By Lemma 3 for d = 2 we have P c ( h, g ′ ) ⊂ M .We intend to show that c is not on the great circle E of S through g and h . In orderto see this, for a while suppose the opposite, i.e. that c ∈ E . Then from | g ′ g | = δ , | g ′ c | = δ and | hc | = δ we conclude that ∠ gg ′ c = ∠ hcg ′ . So the spherical triangle g ′ gc is isosceles,which together with | gg ′ | = δ gives | cg | = δ . Since | gh | = ∆( L ) = ∆( K ) > g is apoint of ch different from c , we get a contradiction. Hence, really, c E .By the preceding paragraph P c ( h, g ′ ) intersects bd( M ) at a point h ′ different from h and g ′ . So the non-empty set P c ( h, g ′ ) \ { h, h ′ } is out of M . This contradicts the result ofthe paragraph before the last. Consequently, K is a body of constant width δ .( ⇐ ) Let us prove that if K is a spherical body of constant width δ , then K is a completebody of diameter δ . In order to prove this, it is sufficient to take any point r K andshow that diam( K ∪ { r } ) > δ .Take the largest ball B ρ ( r ) disjoint with the interior of K . Since K is convex, B ρ ( r ) hasin common with K exactly one point p . By Theorem 3 of [8] there exists a lune L ⊃ K of thickness δ with p as the center of one of the two ( d − q the center of the other ( d − q ∈ K . Since p and q are the centers of the two ( d − L , we have | pq | = δ . From the fact that rp and pq are orthogonalto bd( H ) at p , we see that p ∈ rq . Moreover, p is not an endpoint of rq and | pq | = δ ,Hence | rq | > δ . Thus diam( K ∪ { r } ) > δ . Since r K is arbitrary, K is complete. (cid:3) We say that a convex body D ⊂ S d is of constant diameter δ provided diam( D ) = δ and for every p ∈ bd( D ) there is a point p ′ ∈ bd( D ) with | pp ′ | = δ (see [8]).The following fact is analogous to the result in E d given by Reidemeister [13]. MAREK LASSAK
Theorem 3.
Bodies of constant diameter on S d coincide with complete bodies.Proof. Take a complete body D ⊂ S d of diameter δ . Let g ∈ bd( D ) and G be a hemispheresupporting D at g . By Theorem 2 our D is of constant width δ . So width G (D) = δ and ahemisphere H exists that the lune G ∩ H ⊃ D has thickness δ . By Claim 2 of [6] centers h of H/G and g of G/H belong to D . So | gh | = δ . Thus D is of constant diameter δ .Consider a body D ⊂ S d of constant diameter δ . Let r D . Take the largest B ρ ( r )whose interior is disjoint with D . Denote by p the common point of B ρ ( r ) and D . A uniquehemisphere J supports B ρ ( r ) at p . Observe that D ⊂ J (if not, there is point v ∈ D outof J ; clearly vp passes through int B ρ ( r ), a contradiction). Since D is of constant diameter δ , there is p ′ ∈ D with | pp ′ | = δ . Observe that ∠ rpp ′ ≥ π . If it is π , then | rp ′ | > δ . If it isover π , the triangle rpp ′ is obtuse and then by the law of cosines | rp ′ | > | pp ′ | and hence | rp ′ | > δ . By | rp ′ | > δ in both cases we see that D is complete. (cid:3) By Theorem 2, in Theorem 3 we may exchange “complete” to “constant width”. Thisform is proved earlier as follows. Any body of constant width δ on S d is of constantdiameter δ and the inverse is shown for δ ≥ π , and for δ < π if d = 2 (see [8]). By [4] theinverse holds for any δ . Our short proof of Theorem 3 is different from these in [8] and [4]. References [1] T. Bonnesen, W. Fenchel, Theorie der konvexen K¨orper, Springer, Berlin (1934) (English translation:Theory of Convex Bodies, BCS Associates, Moscow, Idaho, 1987).[2] G. D. Chakerian, H. Groemer, Convex bodies of constant width, In Convexity and its applications,pp. 49–96, Birkh¨auser, Basel (1983).[3] H. G. Eggleston, Convexity, Cambridge University Press, 1958.[4] H. Han, D. Wu, Constant diameter and constant width of spherical convex bodies, arXiv:1905.09098v2.[5] B. Jessen, ¨Uber konvexe Punktmengen konstanter Breite.
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