AA new look at the Blaschke-Leichtweiss theorem ∗ K´aroly Bezdek † January 5, 2021
Abstract
The Blaschke-Leichtweiss theorem (Abh. Math. Sem. Univ. Hamburg 75: 257–284, 2005) statesthat the smallest area convex domain of constant width w in the 2-dimensional spherical space S isthe spherical Reuleaux triangle for all 0 < w ≤ π . In this paper we extend this result to the family ofwide r -disk domains of S , where 0 < r ≤ π . Here a wide r -disk domain is an intersection of sphericaldisks of radius r with centers contained in their intersection. This gives a new and short proof forthe Blaschke-Leichtweiss theorem. Furthermore, we investigate the higher dimensional analogue of wide r -disk domains called wide r -ball bodies. In particular, we determine their minimum spherical width(resp., inradius) in the spherical d -space S d for all d ≥
2. Also, it is shown that any minimum volumewide r -ball body is of constant width r in S d , d ≥ Let S d = { x ∈ E d +1 | (cid:107) x (cid:107) = (cid:112) (cid:104) x , x (cid:105) = 1 } be the unit sphere centered at the origin o in the ( d +1)-dimensionalEuclidean space E d +1 , where (cid:107) · (cid:107) and (cid:104)· , ·(cid:105) denote the canonical Euclidean norm and the canonical innerproduct in E d +1 , d ≥
2. A ( d − S d is an intersection of S d with a hyperplane of E d +1 passing through o (i.e., with a d -dimensional linear subspace of E d +1 ). In particular, an intersection of S d with a 2-dimensional linear subspace in E d +1 is called a great circle of S d . Two points are called antipodesif they can be obtained as an intersection of S d with a line through o in E d +1 . If a , b ∈ S d are two pointsthat are not antipodes, then we label the (uniquely determined) shortest geodesic arc of S d connecting a and b by ab . In other words, ab is the shorter circular arc with endpoints a and b of the great circle (cid:99) ab that passes through a and b . The length of ab is called the spherical distance between a and b and it islabelled by dist s ( a , b ). Clearly, 0 < dist s ( a , b ) < π . If a , b ∈ S d are antipodes, then we set dist s ( a , b ) = π .Let x ∈ S d and r ∈ (0 , π ]. Then the set B ds [ x , r ] := { y ∈ S d | dist s ( x , y ) ≤ r } (resp ., B ds ( x , r ) := { y ∈ S d | dist s ( x , y ) < r } )is called the d -dimensional closed (resp., open) spherical ball, or shorter the d -dimensional closed (resp.,open) ball, centered at x having (spherical) radius r in S d . In particular, B ds [ x , π ] (resp., B ds ( x , π )) is calledthe closed (resp., open) hemisphere of S d with center x . Moreover, B s [ x , r ] (resp., B s ( x , r )) is called theclosed (resp., open) disk with center x and (spherical) radius r in S . Now, the boundary of B ds [ x , r ] (resp., B ds ( x , r )) in S d is called the ( d − S d − s ( x , r ) with center x and (spherical) radius r in S d . As a special case, the boundary of the disk B s [ x , r ] (resp., B s ( x , r )) in S is called the circle with center ∗ Keywords: spherical d -space, convex body of constant width, wide r -disk domain, wide r -ball body, inradius, circumradius,width, volume, Blaschke-Leichtweiss theorem, Jung theorem, isodiametric inequality, Cauchy’s arm lemma.2000 Mathematical Subject Classification. Primary: 52A10, 52A38, 52A55, Secondary: 52A20, 52A40. † Partially supported by a Natural Sciences and Engineering Research Council of Canada Discovery Grant. a r X i v : . [ m a t h . M G ] J a n and of (spherical) radius r and it is labelled by S s ( x , r ). We introduce the following additional notations.For a set X ⊆ S d and r ∈ (0 , π ] let B ds [ X, r ] := (cid:92) x ∈ X B ds [ x , r ] and B ds ( X, r ) := (cid:92) x ∈ X B ds ( x , r ) . Another basic concept is spherical convexity: we say that Q ⊂ S d is spherically convex if it has no antipodesand for any two points x , y ∈ Q we have xy ⊆ Q . (It follows that there exists q ∈ S d such that Q ⊆ B ds ( q , π )with B ds ( x , π ) being spherically convex.) As the intersection of spherically convex sets is spherically convextherefore if X ⊂ B ds ( x , π ), then we define the spherical convex hull conv s X of X as the intersection ofspherically convex sets containing X . By a convex body in S d (resp., a convex domain in S ) we mean aclosed spherically convex set with non-empty interior in S d (resp., in S ). Let K ds , d ≥ S d . If Q ⊆ S d , d ≥
2, then its spherical diameter is diam s ( Q ) := sup { dist s ( x , y ) | x , y ∈ Q } .The core notion of this paper is introduced as follows. Definition 1.1.
Let K ⊂ S d , d ≥ be a closed set with spherical diameter < w := diam s ( K ) ≤ π . Wesay that K is a convex body of constant width w in S d if and only if K = B ds [ K , w ] . Let K ds ( w ) denote thefamily of convex bodies of constant width w in S d for d ≥ and < w ≤ π . Clearly, K ds ( w ) ⊂ K ds for all d ≥ < w ≤ π . On the one hand, the main results (Theorems 1 and 2)proved in [12] imply that Definition 1.1 is in fact, equivalent to the so-called classical definition of sphericalconvex bodies of constant width using normal directions, which is discussed in details in Section 1.3 of [12](see also [23] and in particular, [20] for another equivalent approach). On the other hand, Definition 1.1 helpsto take a new look of the Blaschke-Leichtweiss theorem as discussed in the next two subsections. For the sakeof completeness, we note that a lot more is known about the geometry of convex bodies of constant widthin Euclidean spaces than in spherical (resp., hyperbolic) spaces (see [18], [19], and the recent comprehensivemonograph [24]). In particular, while the Euclidean analogue of Theorem 1.3 of this paper has already beenproved in [7], finding its hyperbolic analogue remains to be seen. The classical Blaschke-Lebesgue theorem states that in the Euclidean plane among all convex sets of constantwidth w (cid:48) > w (cid:48) with centers at the vertices of an equilateral triangle of side length w (cid:48) . For a survey onthis theorem and its impact on extremal geometry we refer the interested reader to the recent elegant papers[18] and [19]. Very different proofs of this theorem were given by Blaschke [8], Lebesgue [22], Fujiwara [15],Eggleston [14], Besicovich [2], Ghandehari [16], Campi, Colesanti, and Gronchi [10], Harrell [17], and M.Bezdek [7]. So, it is natural to ask whether any of these proofs can be extended to the spherical plane.Actually, Blaschke claimed that this can be done with his Euclidean proof (see [8], p. 505), but one had towait until Leichtweiss did it (using some ideas of Blaschke) in [23]. So, we call the following statement theBlaschke-Leichtweiss theorem: if K ∈ K s ( w ) with 0 < w ≤ π , thenarea s ( K ) ≥ area s ( ∆ ( w )) , (1)where area s ( · ) refers to the spherical area of the corresponding set in S and ∆ ( w ) denotes the sphericalReuleaux triangle which is an intersection of three disks of radius w with centers at the vertices of a sphericalequilateral triangle of side length w . As the only known proof of (1) is the one published in [23] which isa combination of geometric and analytic ideas presented on twenty pages, one might wonder whether thereis a simpler approach. This paper intends to fill this gap by proving a stronger result (Theorem 1.3) in anew and shorter way. The Euclidean analogue of Theorem 1.3 has already been proved in [7] and our proofof Theorem 1.3 presented below is an extension of the Euclidean technique of [7] to S combined with theproperly modified spherical method of [3]. For the sake of completeness we note that in [3] the author andBlekherman proved a spherical analogue of P´al’s theorem stating that the minimal spherical area convexdomain of given minimal spherical width ω is a regular spherical triangle for all 0 < ω ≤ π .2 .3 The Blaschke-Leichtweiss theorem extended: minimizing the area of wide r -disk domains in S The following definition introduces wide r -disk domains in S the spherical areas of which we wish to minimizefor given 0 < r ≤ π . Definition 1.2.
Let < r ≤ π be given and let ∅ (cid:54) = X be a closed subset of S with diam s ( X ) ≤ r . Then B s [ X, r ] is called the wide r -disk domain generated by X in S . The family of wide r -disk domains of S islabelled by B s, wide ( r ) . Now, (1) can be generalized as follows.
Theorem 1.3.
Let < r ≤ π and D ∈ B s, wide ( r ) . Then area s ( D ) ≥ area s ( ∆ ( r )) . As ∆ ( r ) ∈ K s ( r ) ⊂ B s, wide ( r ) holds for all 0 < r ≤ π therefore (1) follows from Theorem 1.3 in astraightforward way. Moreover, we note that our method for proving Theorem 1.3 is completely differentfrom the ideas and techniques used in [23] and so, in this way we obtain a new proof of the Blaschke-Leichtweiss theorem, which turns out to be much shorter than the one published in [23].The rest of the paper is organized as follows. Section 3 gives a proof of Theorem 1.3 via successive areadecreasing cuts and symmetrization. On the other hand, that proof is based on some extremal propertiesof wide r -disk domains, which are proved in Section 2. Furthermore, Section 2 investigates the higherdimensional analogue of wide r -disk domains called wide r -ball bodies for 0 < r ≤ π . In particular, wedetermine their minimum spherical width (resp., inradius) in S d , d ≥
2. Also, it is shown that any minimumvolume wide r -ball body is of constant width r in S d , d ≥ r -ballbodies in S d for d ≥ and < r ≤ π S d thereby introducing the family of wide r -ball bodies (resp., wide r -ball polyhedra) in S d as follows. Definition 2.1.
Let < r ≤ π be given and let ∅ (cid:54) = X be a closed subset of S d , d ≥ with diam s ( X ) ≤ r .Then B ds [ X, r ] is called a wide r -ball body generated by X in S d . The family of wide r -ball bodies of S d islabelled by B ds, wide ( r ) . If X ⊂ S d with < card( X ) < + ∞ and diam s ( X ) ≤ r , then B ds [ X, r ] is called a wide r -ball polyhedron generated by X in S d . The family of wide r -ball polyhedra of S d is labelled by P ds, wide ( r ) . We leave the straightforward proof of the following claim (using Definition 2.1) to the reader.
Proposition 2.2.
Every wide r -ball body B ds [ X, r ] ∈ B ds, wide ( r ) , d ≥ , < r ≤ π can be approximated(in the Hausdorff sense) arbitrarily close by a suitable wide r -ball polyhedron and therefore there exists asequence P n ∈ P ds, wide ( r ) , n = 1 , , . . . such that lim n → + ∞ vol s ( P n ) = vol s ( B ds [ X, r ]) , where vol s ( · ) standsfor the d -dimensional spherical volume of the corresponding set in S d . Although the question of finding an extension of Theorem 1.3 to S d for d ≥ Problem 2.3.
Find c BL ( r, d ) := inf { vol s ( B ds [ X, r ]) | B ds [ X, r ] ∈ B ds, wide ( r ) } = inf { vol s ( P ds [ X, r ]) | P ds [ X, r ] ∈ P ds, wide ( r ) } for given < r ≤ π and d ≥ . Proposition 2.5 can be used to lower bound c BL ( r, d ) with the spherical volume of a properly chosen ball.3 efinition 2.4. The smallest ball (resp., the largest ball) containing (resp., contained in) the convex body K ∈ K ds , d ≥ is called the circumscribed (resp., inscribed) ball of K whose radius R cr ( K ) (resp., R in ( K ) )is called the circumradius (resp., inradius) of K . Proposition 2.5.
Let B ds [ X, r ] ∈ B ds, wide ( r ) with d ≥ and < r ≤ π . Then R in ( B ds [ X, r ]) ≥ R in ( ∆ d ( r )) , where ∆ d ( r ) ∈ B ds, wide ( r ) denotes the intersection of d + 1 closed balls of radii r centered at the vertices of aregular spherical d -simplex of edge length r in S d .Proof. As B ds [ X, r ] ∈ B ds, wide ( r ) therefore diam s ( X ) ≤ r . This and the spherical Jung theorem [13] implythat there exists x ∈ S d such that X ⊂ B ds [ x , R cr ( ∆ d ( r ))]. It follows that B ds [ x , R in ( ∆ d ( r ))] = B ds [ x , r − R cr ( ∆ d ( r ))] ⊂ B ds [ X, r ]and therefore R in ( B ds [ X, r ]) ≥ R in ( ∆ d ( r )), finishing the proof of Proposition 2.5.For more details on the concepts introduced in Definitions 2.6, 2.7, and 2.8, we refer the interested readerto the recent paper of Lassak [20]. As usual, we say that the ( d − C d − s ( x , π ) isa supporting ( d − K ∈ K ds if C d − s ( x , π ) ∩ K (cid:54) = ∅ and K ⊂ B ds [ x , π ], in whichcase B ds [ x , π ] is called a closed supporting hemisphere of K . One can show that through each boundarypoint of K there exists at least one supporting ( d − K moreover, K is theintersection of its closed supporting hemispheres. Two hemispheres of S d are called opposite if the theircenters are antipodes. Definition 2.6.
The intersection of two distinct closed hemispheres of S d which are not opposite is called alune of S d . Let L ds denote the family of lunes in S d . Every lune of S d is bounded by two ( d − d − S d ) sharing a pair of antipodes in common, which are called the vertices of the lune. Definition 2.7.
The angular measure of the angle formed by the two ( d − -dimensional hemispheresbounding the lune L ∈ L ds (which is equal to spherical distance of the centers of the two ( d − -dimensionalhemispheres bounding L ) is called the spherical width of the given lune labelled by width s ( L ) . Definition 2.8.
For every closed supporting hemisphere H of the convex body K ∈ K ds there exists a closedsupporting hemisphere H (cid:48) of K such that the lune H ∩ H (cid:48) has minimal width for given H and K . We call width s ( H ∩ H (cid:48) ) the width of K determined by H and label it by width H ( K ) . Finally, the minimal sphericalwidth (also called thickness) width s ( K ) of K is the smallest spherical width of the lunes that contain K , i.e., width s ( K ) = min { width H ( K ) | H is a closed supporting hemisphere of K } . Next, we recall the following claim from [20] (Claim 2), which is often applicable.
Sublemma 2.9.
Let K ∈ K ds . If L ∈ L ds contains K and width s ( K ) = width s ( L ) , then both centers of the ( d − -dimensional hemispheres bounding L belong to K . Remark 2.10.
We note that Definition 2.8 supports to say that the convex body K ∈ K ds is of constant width < w ≤ π in S d if the width of K with respect to any supporting hemisphere is equal to w . Theorem 2 of [21]proves that this definition of constant width is equivalent to the one under Definition 1.1, i.e., K ∈ K ds ( w ) for d ≥ and < w ≤ π if and only if width s ( K ) = diam s ( K ) = w holds for K ∈ K ds . Now, we are ready to prove the following close relative of Proposition 2.5.
Proposition 2.11.
Let B ds [ X, r ] ∈ B ds, wide ( r ) with d ≥ and < r ≤ π . Then width s ( B ds [ X, r ]) ≥ width s ( ∆ d ( r )) = r. roof. It will be convenient to use the following notion (resp., notation) from [6].
Definition 2.12.
For a set X ⊆ S d , d ≥ and < r ≤ π let the r -dual set X r of X be defined by X r := B ds [ X, r ] . If the spherical interior int s ( X r ) (cid:54) = ∅ , then we call X r the r -dual body of X . r -dual sets satisfy some basic identities such as (( X r ) r ) r = X r and ( X ∪ Y ) r = X r ∩ Y r , which hold for any X ⊆ S d and Y ⊆ S d . Clearly, also monotonicity holds namely, X ⊆ Y ⊆ S d implies Y r ⊆ X r . Thus, thereis a good deal of similarity between r -dual sets and spherical polar sets in S d . For more details see [6]. Thefollowing statement is a spherical analogue of Lemma 3.1 in [4]. Sublemma 2.13.
Let H be a closed supporting hemisphere of the r -dual body X r of X ⊂ S d boundedby the ( d − -dimensional great sphere H in S d such that H and H support X r at the boundary point x ∈ H ∩ bd( X r ) , where d ≥ , and < r ≤ π . Then the d -dimensional closed ball of radius r of S d that istangent to H at x and lies in H contains the r -dual body X r .Proof. (The following proof is the spherical analogue of the Euclidean proof of Lemma 3.1 of [4].) Let B ds [ c , r ]be the d -dimensional closed ball of radius r of S d that is tangent to H at x and lies in H . Assume that X r is not contained in B ds [ c , r ], i.e., let y ∈ X r \ B ds [ c , r ]. Then by taking the intersection of the configurationwith the 2-dimensional spherical plane spanned by x , y , and c we see that there is a shorter circular arc ofradius r connecting x and y that is not contained in B ds [ c , r ] and therefore it is not supported by neither H nor H . On the other hand, as x , y ∈ X r therefore any such arc must be contained in X r and must besupported by H as well as H , a contradiction.Now, let X r = B ds [ X, r ] ∈ B ds, wide ( r ) with d ≥ < r ≤ π . Sublemma 2.9 implies that thereexists L ∈ L ds such that X r ⊆ L := B ds [ x , π ] ∩ B ds [ y , π ] and x (cid:48) ∈ S d − s ( x , π ) ∩ X r is the center of the( d − S d − s ( x , π ) ∩ B ds [ y , π ] and y (cid:48) ∈ S d − s ( y , π ) ∩ X r is the center of the ( d − S d − s ( y , π ) ∩ B ds [ x , π ] satisfying width s ( X r ) = width s ( L ) = dist s ( x (cid:48) , y (cid:48) ). It followsfrom Sublemma 2.13 in a straightforward way that there exists B ds [ x (cid:48)(cid:48) , r ] (resp., B ds [ y (cid:48)(cid:48) , r ]) such that X r ⊆ B ds [ x (cid:48)(cid:48) , r ] ⊆ B ds [ x , π ] (resp., X r ⊆ B ds [ y (cid:48)(cid:48) , r ] ⊆ B ds [ y , π ]) and B ds [ x (cid:48)(cid:48) , r ] (resp., B ds [ y (cid:48)(cid:48) , r ]) is tangent to S d − s ( x , π ) (resp., S d − s ( y , π )) at x (cid:48) (resp., y (cid:48) ) with x (cid:48)(cid:48) ∈ ( X r ) r (resp., y (cid:48)(cid:48) ∈ ( X r ) r ). By construction2 r − width s ( X r ) = 2 r − dist s ( x (cid:48) , y (cid:48) ) = dist s ( x (cid:48)(cid:48) , y (cid:48)(cid:48) ) ≤ diam s (( X r ) r )and therefore 2 r ≤ width s ( X r ) + diam s (( X r ) r ) . (2) Sublemma 2.14.
Let < r ≤ π be given and let ∅ (cid:54) = X be a closed subset of S d , d ≥ with diam s ( X ) ≤ r .Then diam s (( X r ) r ) ≤ r. (3) Proof.
Recall ([12] or [21]) that a closed set Y ⊂ S d is called a complete set if diam s ( Y ∪ { y } ) > diam s ( Y )holds for all y ∈ S d \ Y . It is easy to prove the following claim (see Lemma 1 of [21]): if Y is a complete setwith diam( Y ) ≤ π , then Y = Y diam s ( Y ) ∈ K ds (diam s ( Y )) ⊂ K ds . Furthermore, it is well know (see Theorem1 of [12] or Theorem 1 of [21]) that each set of diameter δ ∈ (0 , π ) in S d is a subset of a complete set ofdiameter δ in S d . Thus, there exists a complete set Y ⊂ S d such that X ⊆ Y with diam s ( X ) ≤ diam s ( Y ) = r .By the monotonicity of the r -dual operation it follows that ( X r ) r ⊆ ( Y r ) r = Y and so, diam s (( X r ) r ) ≤ diam s ( Y ) = r .Thus, (2) and (3) yield r ≤ width s ( X r ), finishing the proof of Proposition 2.11.In fact, c BL ( r, d ) is equal to the minimum of the volumes of convex bodies of constant width r in S d asstated in Proposition 2.15. This can be proved as follows. Let 0 < r ≤ π be given and let ∅ (cid:54) = X be aclosed subset of S d , d ≥ s ( X ) ≤ r . Then the proof of Sublemma 2.14 shows the existence of acomplete set Y ⊂ S d such that X ⊆ Y with diam s ( X ) ≤ diam s ( Y ) = r . As Y r = Y therefore Y is a convexbody of constant width r , i.e., Y ∈ K ds ( r ). Moreover, the monotonicity of the r -dual operation implies that5 = Y r ⊆ X r , where X r = B ds [ X, r ] ∈ B ds, wide ( r ). Finally, Blaschke’s selection theorem applied to K ds ( r )([25]) yields Proposition 2.15.
Every wide r -ball-body B ds [ X, r ] ∈ B ds, wide ( r ) contains a convex body of constant width r ,i.e., there exists Y = B ds [ Y, r ] ∈ K ds ( r ) such that Y ⊆ B ds [ X, r ] , where < r ≤ π and d ≥ . Thus, c BL ( r, d ) = min { vol s ( K ) | K ∈ K ds ( r ) } holds for all < r ≤ π and d ≥ . In connection with Proposition 2.15 it is natural to look for the spherical analogue of Schramm’s lowerbound ([26]) for the volume of convex bodies of constant width in E d . This has been done by Schramm ([27])for c BL (cid:0) π , d (cid:1) = min (cid:8) vol( K ) | K ∈ K ds (cid:0) π (cid:1)(cid:9) as follows. Remark 2.16.
Proposition 9 of [27] implies that c BL (cid:16) π , d (cid:17) ≥ (cid:115) d π ( d + 1)( d + 4) d vol s (cid:16) ∆ d (cid:16) π (cid:17)(cid:17) , where vol s ( S d ) = ( d + 1) ω d +1 = ( d +1) π d +12 Γ( d +32 ) , vol s (cid:0) ∆ d (cid:0) π (cid:1)(cid:1) = ( d +1) ω d +1 d +1 and d ≥ . It seems reasonable to hope for the following strengthening of the estimate of Remark 2.16 (resp., ofConjecture 1.6 from [5]).
Conjecture 2.17. c BL ( π , d ) = vol s (∆ d ( π )) , i.e., if K ∈ K ds ( π ) , then vol s ( K ) ≥ vol s (∆ d ( π )) for all d ≥ . Let 0 < r ≤ π and D ∈ B s, wide ( r ). Our goal is to show that area s ( D ) ≥ area s ( ∆ ( r )). Let C in be theinscribed disk of D with center c having radius R in . We may assume that 2 R in < r . Namely, if r ≤ R in ,then D contains a disk of diameter r and so, it follows via the spherical isodiametric inequality ([9]) thatarea s ( D ) ≥ area s ( ∆ ( r )).Next, one of the following two cases must occur: either the boundary of D and C in have two points incommon such that the shorter great circular arc connecting them is a diameter of C in or the boundary of D and C in have three points in common such that c is in the interior of the triangle that is the sphericalconvex hull of these three points. In the first case, Sublemma 2.13 implies that D contains a disk of diameter r and so, as above we get that area s ( D ) ≥ area s ( ∆ ( r )).In the second case, let the three selected points in common of the boundary of D and C in be a , a , and a and let the supporting great circles to C in at these points be L , L , and L respectively (Figure 1). Notethat L , L , and L are also supporting great circles to D and thus, D lies in one of the spherical trianglesdetermined by L , L , and L . Let us label this spherical triangle by (cid:52) n n n having the vertices n , n ,and n such that that n is not on L (i.e., n is “opposite” to L ), n is not on L , and n is not on L .Now, let p be the point on the same side of L as D such that p a is of length r and is perpendicular to L at a . Let M be the great circle perpendicular to p a at p . Note that the angle between L and M is r . By Proposition 2.11, M must contain a point of D . Let this point be q . Since 0 < r ≤ π we havethat dist s ( q , c ) ≥ dist s ( p , c ) = r − R in > R in . (4)Let t and t be the two points in common of the boundary of C in with the two circles of radius r passing through q that are tangent to C in and whose disks of radius r contain C in . Here the shortercircular arc of radius r connecting q and t (resp., t ) and sitting on the corresponding circle of radius r just introduced, is labeled by ( q t ) r (resp., ( q t ) r ). The circular arcs ( q t ) r and ( q t ) r have equallengths moreover, the cap C bounded by ( q t ) r and ( q t ) r and the shorter circular arc of radius R in C of the cap-domain C := C ∪ C ∪ C ∪ C in in the hemisphere of S with center c .connecting t and t on the boundary of C in lies in D and therefore it lies also in the spherical triangle (cid:52) n a a ⊂ (cid:52) n n n . (Here we have used the property of D that if we choose two points in D , then anyshorter circular arc of radius r (cid:48) with r ≤ r (cid:48) ≤ π connecting the two points lies in D .) We can perform thesame procedure for the points a and a , producing the caps C and C . By construction the caps C , C ,and C are non-overlapping and the cap-domain C := C ∪ C ∪ C ∪ C in is a subset of D and thereforearea s ( D ) ≥ area s ( C ) . (5)Let D ∗ := ∆ ( r ) with vertices b , b , and b such that its center is c (Figure 2). If R ∗ in denotes theinradius of D ∗ , then dist s ( b , c ) = dist s ( b , c ) = dist s ( b , c ) = r − R ∗ in . Clearly, Proposition 2.5 yields that r − R ∗ in ≥ r − R in . Thus, let c i be the point on the great circular arc b i c such that dist s ( c i , c ) = r − R in ,1 ≤ i ≤
3. Let t ∗ and t ∗ be the two points in common of the boundary of C in with the two circles of radius r passing through c that are tangent to C in and whose disks of radius r contain C in . Here the shorter circulararc of radius r connecting c and t ∗ (resp., t ∗ ) and sitting on the corresponding circle of radius r justintroduced, is labeled by ( c t ∗ ) r (resp., ( c t ∗ ) r ). The circular arcs ( c t ∗ ) r and ( c t ∗ ) r have equal lengths.Moreover, let the cap C ∗ be the domain bounded by ( c t ∗ ) r and ( c t ∗ ) r and the shorter circular arc ofradius R in connecting t ∗ and t ∗ on the boundary of C in . From (4) it follows that area s ( C ) ≥ area s ( C ∗ ).Similarly, we can define the caps C ∗ and C ∗ with vertices c and c for which area s ( C ) ≥ area s ( C ∗ ) andarea s ( C ) ≥ area s ( C ∗ ). By construction the caps C ∗ , C ∗ , and C ∗ are non-overlapping and therefore thecap-domain C ∗ := C ∗ ∪ C ∗ ∪ C ∗ ∪ C in satisfies the inequalityarea s ( C ) ≥ area s ( C ∗ ) . (6)Based on (5) and (6) we finish the proof of Theorem 1.3 by showing the inequalityarea s ( C ∗ ) ≥ area s ( D ∗ ) . (7)Let b be the midpoint of ( b b ) r , which is the shorter circular arc of radius r connecting b and b on the boundary of D ∗ (Figure 2). Moreover, let b ∗ := (cid:100) cb ∩ ( S s ( c , R in ) \ cb ). From this it follows that7igure 2: The cap-domain C ∗ := C ∗ ∪ C ∗ ∪ C ∗ ∪ C in compared to D ∗ via dissection and symmetry.dist s ( b , c ) = dist s ( b , b ∗ ) = R in − R ∗ in . Let f := ( b b ) r ∩ ( c b ∗ ) r , where ( b b ) r (resp., ( c b ∗ ) r ) isthe shorter circular arc of radius r connecting b and b (resp., c and b ∗ ) such that ( b b ) r lies on theboundary of D ∗ (resp., the disk B s [ c (cid:48) , r ] containing ( c b ∗ ) r on its boundary contains b (resp., b ) in itsinterior (resp., exterior)). We note that by construction( c b ∗ ) r ⊂ C ∗ . (8) Sublemma 3.1.
Let u := (cid:100) b c (cid:48) ∩ ( S s ( b , r ) \ B s [ c (cid:48) , r ]) and v := (cid:100) b c (cid:48) ∩ ( S s ( c (cid:48) , r ) \ B s [ b , r ]) (Figure 3).Furthermore, let ( fv ) r be the shorter circular arc of S s ( c (cid:48) , r ) connecting f and v and let x ∈ ( fv ) r be a pointmoving from f to v . Then the point of B s [ b , r ]) closest to x is y := b x ∩ S s ( b , r ) and dist s ( x , y ) is astrictly increasing function of the length of ( fx ) r , where ( fx ) r denotes the shorter circular arc of S s ( c (cid:48) , r ) connecting f and x .Proof. Clearly, the point of B s [ b , r ] closest to x must have the property that the great circle passing throughit and tangent to B s [ b , r ] is orthogonal to the great circular arc connecting that point to x . It follows thatthe closest point is y = b x ∩ S s ( b , r ). On the other hand, notice that as x ∈ ( fv ) r moves from f to v the angle ∠ b c (cid:48) x at the vertex c (cid:48) of the spherical triangle (cid:52) b c (cid:48) x (bounded by the great circular arcs b c (cid:48) , c (cid:48) x and b x ) strictly increases and so, the spherical version of Cauchy’s Arm Lemma (see [1] or [11],p. 228) implies that dist s ( b , x ) strictly increases and therefore also dist s ( x , y ) = dist s ( b , x ) − r strictlyincreases.Next, we note that the spherical distace of b ∗ (resp., b ) to B s [ b , r ] (resp., B s [ c (cid:48) , r ]) is equal todist s ( b ∗ , b ) = R in − R ∗ in (resp., is at most dist s ( b , c ) = R in − R ∗ in ). Hence, Sublemma 3.1 implies thatthe length of ( b f ) r (resp., ( c f ) r ) is at most as large as the length of ( b ∗ f ) r (resp., ( b f ) r ), where ( b f ) r ,( c f ) r , ( b ∗ f ) r , and ( b f ) r are circular arcs of radius r with endpoints indicated such that ( b f ) r ⊂ ( b b ) r ,( c f ) r ⊂ ( c b ∗ ) r , ( b ∗ f ) r ⊂ ( c b ∗ ) r , and ( b f ) r ⊂ ( b b ) r . It follows that the triangular shape region8igure 3: Cauchy’s Arm Lemma applied to (cid:52) b c (cid:48) x in S (cid:98) (cid:52) b c f bounded by ( b f ) r , b c , and ( c f ) r has an isometric copy contained in the triangle shape region (cid:98) (cid:52) b ∗ b f bounded by ( b ∗ f ) r , b ∗ b , and ( b f ) r . This implies thatarea s ( (cid:98) (cid:52) b c f ) ≤ area s ( (cid:98) (cid:52) b ∗ b f ) . (9)Thus, using (8), (9), and the symmetries of D ∗ and C ∗ we get that16 area s ( D ∗ \ C ∗ ) ≤ area s ( (cid:98) (cid:52) b c f ) ≤ area s ( (cid:98) (cid:52) b ∗ b f ) ≤
16 area s ( C ∗ \ D ∗ ) . (10)Hence, (7) follows, finishing the proof of Theorem 1.3. References [1] Z. Abel, D. Charlton, S. Collette, E.D. Demaine, M. L. Demaine, S. Langerman, J. O’Rourke, V.Pinciu, and G. Toussaint,
Cauchy’s Arm Lemma on a growing sphere , arXiv:0804.0986v1 [cs.CG](2008), 1–10.[2] A. S. Besicovich,
Minimum area of a set of constant width , Proc. Symp. Pure Math. (1963), 13–14.[3] K. Bezdek and G. Blekherman, Danzer-Gr¨unbaum’s theorem revisited , Periodica Math. Hungar. (1999), 7–15.[4] K. Bezdek, Zs. L´angi, M. Nasz´odi, and P. Papez,
Ball-polyhedra , Discrete Comput. Geom. (2007), 201–230.[5] K. Bezdek,
Illuminating spindle convex bodies and minimizing the volume of spherical sets of constantwidth , Discrete Comput. Geom. (2012), 275–287.[6] K. Bezdek,
From r-dual sets to uniform contractions , Aequationes Math. (2018), 123–134.[7] M. Bezdek,
On a generalization of the Blaschke-Lebesgue theorem for disk-polygons , Contrib. DiscreteMath. (2011), 77–85. 98] W. Blaschke,
Konvexe Bereiche gegebener konstanter Breite und kleinsten Inhalts , Math. Ann. (1915), 504–513.[9] K. J. B¨or¨oczky and ´A. Sagmeister, The isodiametric problem on the sphere and in the hyperbolicspace , Acta Math. Hungar. (2020), 13–32.[10] S. Campi, A. Colesanti and P. Gronchi,
Minimum problems for volumes of constant bodies , in:Partial Differential Equations and Applications, Eds.: P. Marcellini, G. Talenti, and E. Visintin,Marcel-Dekker, New York (1996), 43–55.[11] P. Cromwell,
Polyhedra , Cambridge University Press, 1997.[12] B. V. Dekster,
Completeness and constant width in spherical and hyperbolic spaces , Acta Math.Hungar. (1995), 289–300.[13] B. V. Dekster,
The Jung theorem for spherical and hyperbolic spaces , Acta Math. Hungar. (1995), 315–331.[14] H. G. Eggleston,
A proof of Blaschke’s theorem on the Reuleaux triangle , Quart. J. Math. Oxford (1952), 296–297.[15] M. Fujiwara, Analytical proof of Blaschke theorem on the curve of constant breadth with minimumarea I and II , Proc. Imp. Acad. Japan (1927), 307–309 and (1931), 300–302.[16] M. Ghandehari, An optimal control formulation of the Blaschke-Lebesgue Theorem , J. Math. Anal.Appl. (1996), 322–331.[17] E. M. Harrell II,
A direct proof of a theorem of Blaschke and Lebesgue , J. Geom. Anal. (2002),81–88.[18] B. Kawohl, Convex sets of constant width , Oberwolfach Reports (2009), 390–393.[19] B. Kawohl and C. Weber, Meissner’s mysterious bodies , Math. Intelligencer (2011), 94–101.[20] M. Lassak,
Width of spherical convex bodies , Aequationes Math. (2015), 555–567.[21] M. Lassak,
Complete spherical convex bodies , J. Geom. (2020), Paper No. 35, 6 pp.[22] H. Lebesgue,
Sur le problme des isoprimtres et sur les domaines de largeur constante , Bull. Soc.Math. France C.R. (1914), 72–76.[23] K. Leichtweiss, Curves of constant width in the non-Euclidean geometry , Abh. Math. Sem. Univ.Hamburg (2005), 257–284.[24] H. Martini, L. Montejano, and D. Oliveros, Bodies of constant width - An introduction to convexgeometry with applications , Birkh¨auser/Springer, Cham, 2019.[25] R. Schneider,
Convex Bodies: The Brunn-Minkowski Theory , 2nd edn. Encyclopedia of Mathematicsand its Applications, vol. 151, Cambridge University Press, Cambridge, 2014.[26] O. Schramm,
On the volume of sets having constant width , Israel J. Math. (1988), 178–182.[27] O. Schramm,
Illuminating sets of constant width , Mathematika (1988), 180–189.K´aroly Bezdek Department of Mathematics and Statistics, University of Calgary, Calgary, CanadaDepartment of Mathematics, University of Pannonia, Veszpr´em, Hungary [email protected]@math.ucalgary.ca