Issue 
Wuhan Univ. J. Nat. Sci.
Volume 27, Number 5, October 2022



Page(s)  367  371  
DOI  https://doi.org/10.1051/wujns/2022275367  
Published online  11 November 2022 
Mathematics
CLC number: O 186
The Minkowski Measure of Asymmetry for Spherical Bodies of Constant Width
School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou 215009, Jiangsu, China
^{†} To whom correspondence should be addressed. Email: jinhailin17@163.com
Received:
10
May
2022
In this paper, we introduce the Minkowski measure of asymmetry for the spherical bodies of constant width. Then we prove that the spherical balls are the most symmetric bodies among all spherical bodies of constant width, and the completion of the spherical regular simplexes are the most asymmetric bodies.
Key words: spherical convex body / spherical body of constant width / Minkowski measure of asymmetry / simplex / Reuleaux triangle
Biography: HOU Peiwen, female, Master candidate, research direction: convex geometric analysis. Email: houpeiwen6@163.com
Fundation item: Supported by the National Natural Science Foundation of China (12071334, 12071277)
© Wuhan University 2022
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
0 Introduction
The Minkowski measure of asymmetry of convex bodies in the Euclidean space was introduced by Minkowski^{[1]}. It is wellknown that the dimensional simplex has the biggest Minkowski measure of asymmetry. In other words, we always say that the simplex is the most asymmetric convex body. There are many other measures of asymmetry for convex bodies, such as the Winternitz measure of symmetry, the KovnerBesicovitch measure of symmetry, and the mean Minkowski measure of asymmetry. For the new development of the research of the Minkowski measure of asymmetry, see Refs.[27].
The Convex bodies of constant width are important ones in . The symmetry of convex bodies of constant width in was studied by Besicovitch^{[8]}, who showed that the Reuleaux triangles are the most asymmetric convex bodies of constant width in , and the circles are the most symmetric convex bodies of constant width in . A special measure of asymmetry for convex bodies of constant width in was introduced by Groemer and Wallen^{[9]}. The Minkowski measure of asymmetry for convex bodies of constant width was studied by Guo and Jin^{[3, 5, 6, 10, 11]}. In the sense of Minkowski measure of asymmetry, the complete bodies of the regular simplex are the most asymmetric convex bodies of constant width, and the Euclidean balls are the most symmetric convex bodies of constant width.
The convex geometry in the ndimensional spherical space was investigated by many mathematician, such as Robinson, Santaló, Dekster, Leichtweiss^{[12 16]}, Lassak^{[17]} and Guo^{[10]}. In this paper, we introduce the Minkowski measure of asymmetry for spherical convex bodies of constant width in . Then we prove a spherical analogy of a result obtained by Jin and Guo^{[11, 18]}, i.e. that the spherical balls are the most symmetric bodies among all spherical bodies of constant width, and the completions of the spherical regular simplex are the most asymmetric bodies. Concretely, we give the following theorem.
Main Theorem Let be a spherical body of constant width. Then the Minkowski measure of asymmetry of satisfies the following inequality:
The equality holds on the lefthand side if and only if is a spherical ball, and on the righthand side if and only if is a completion of a spherical regular simplex.
1 Preliminaries
Let denote the class of convex bodies (compact convex sets with nonempty interiors) in , and let denote the class of sets in which contain the origin in their interiors. The convex body is said to be of constant width if its projection on any straight line is a segment of length , which is equivalent to the geometrical fact that any two parallel support hyperplanes of are always at the distance . The convex bodies of constant width in and are also called orbiforms and spheroforms,respectively. Euclidean balls are obviously bodies of constant width, however, there are many others^{[19]}. We denote by the set of all dimensional convex bodies of constant width.
Convex bodies of constant width have many interesting properties and applications which have gained much attention in the history, e.g., orbiforms were intensely studied during the nineteenth century and later, particularly by Reuleaux, whose name is now attached to the orbiforms obtained by intersecting a finite number of disks of equal radii. In , Meissner tetrahedrons may be the most famous spheroforms. Mathematicians believe that Meissner tetrahedrons have the minimal volume among all spheroforms of the same width.
Given a convex body and . For a hyperplane through and the pair of support hyperplanes , (of ) parallel to , let be the ratio, not less than 1, in which divides the distance between and . Put
and define the Minkowski measure of asymmetry of by
A point satisfying is called a critical point of C. The set of all critical points of is denoted by . It is known that is a nonempty convex set.
If , then
Equality holds on the lefthand side if and only if is centrally symmetric, and on the righthand side if and only if is a simplex.
The Minkowski measure of asymmetry for convex bodies of constant width was studied by Jin and Guo^{[11, 18]}.
The critical set of is a singleton, and the unique critical point of is the center of circumscribed sphere of , also the center of inscribed sphere of . Denoted by and be the radii of insphere and circumsphere of , respectively. For , we have . By Jung's theorem, we have
The equality holds on the lefthand side if and only if is an Euclidean ball, and on the righthand side if and only if is a completion of a regular simplex.
In , a completion of a convex body is a convex body of constant width, which has the same diameter as .
2 Spherical Bodies of Constant Width
Let be the unit sphere of the dimensional Euclidean space , where . The intersection of with an dimensional subspace of , where , is called an dimensional subsphereof . In particular, if , we get the 0dimensional subsphere consisting of a pair of antipodal points, and if we obtain the socalled great circle.
If are not antipodes, by the arc connecting them we mean the shorter part of the great circle containing and . By the spherical distance, or distance in short, of these points we know the length of the arc connecting them. Clearly , the angel between vectors , , where is the center of . The diameterdiam() of a set is the supremum of the spherical distances between pairs of points in .
By a spherical ball of radius, or an sballin short, we understand the set of points of having distance at most from a fixed point, called the center of this ball. Spherical balls of radius are called hemispheres. In other words, by a hemisphere of we mean the common part of with any closed halfspace of . We denote by the hemisphere whose center is . Two hemispheres whose centers are antipodes are called opposite hemispheres.
By a sphericaldimensional ball of radius we mean the set of points of a dimensional sphere of which are at distance at most from a fixed point, called the center of this ball. The dimensional balls of radius are called dimensional hemisphere. If , we call them semicircles.
We say that a set is sconvex if it does not contain any pair of antipodes, and if together with for every two points of , the whole arc connecting them is a subset of . By an sconvex body on we mean a closed convex set with nonempty interior.
If an dimensional sphere of has a common point with a convex body and its intersection with the interior of is empty, we say that is a supporting hypersphere ofpassing through. We also say that supports at . If is the hemisphere bounded by and containing , we say that is a supporting hemisphere of and that supports at .
For any distinct nonopposite hemispheres and the set is called a lune of . The dimensional hemispheres bounding the lune which are contained in and , respectively, are denoted by and . By the thickness of a lune we mean the spherical distance of the centers of the dimensional hemispheres and bounding ^{[20]}.
We say that a lune passes through a boundary point of a convex body if the lune contains and if the boundary of the lune contains . If the centers of both dimensional hemispheres bounding a lune belong to , then we call such a lune an orthogonally supporting lune of .
For an sconvex body and any hemisphere supporting we define the width of determined by , denoted by , as the minimum thickness of a lune over all hemispheres supporting ^{[20]}. By a compactness argument we see that at least one such a hemisphere exists, and thus at least one corresponding lune exists. We say that a convex body is of constant width provided for every supporting hemisphere of we have ^{[20]}.
Lemma 1^{[17, 20]} Every two convex sets on with disjoint interiors are subsets of two oppositehemispheres.
Lemma 2^{[17]}If is a body of constant width , then .
We say that a convex body with diameter is of constant diameter provided that for arbitrary there exists such that ^{[20, 21]}. We say that any subset of a hemisphere of which is the largest (in the sense of inclusion) set of a given diameter is acomplete set of diameter , or a complete set for brevity.
Lemma 3^{[17]}
(i) Bodies of constant diameter oncoincide with complete bodies;
(ii) Bodies of constant diameter on coincide with bodies of constant width.
3 The Circumscribed Ball of Spherical Body of Constant Width
In the dimensional Euclidean space , if is a convex bodies of constant width , then the insphere and circumsphere are concentric and their radii, and , respectively, satisfy
and
In this section, we give a similar result of spherical bodies of constant width to the one for convex bodies in Euclidean spaces. The following results and approach of their proofs are implied vaguely in Ref.[22].
For and , denote , the spherical ball with radius and centered at .
Lemma 4 Letbe an sconvex body with diameter. If, then .
Proof Suppose does not contain . Let and let be the great circle through and . Take the point being the intersection of with such that is inbetween and , i.e. locates on the short arc connecting and , and take such that is inbetween and . Then and hence , contradicting the fact that the diameter of is .
Lemma 5 Letbe an sconvex body of constant width. If, then .
Proof Suppose there is a point of that is not in . Let be the great circle through and . By Lemma 1, there exist two opposite hemispheres and such that , and is perpendicular to . Let , be the two intersecting points of with the boundary of , such that is inbetween and . Let be a point of the intersection of with such that is inbetween and . Let be the unique hemisphere supports at . Then the lune contains . Notice that the thickness of the lune is less than , which implies that is not a spherical body of constant width.
Theorem 1 Letbe an sconvex body of constant width. Then the insphere and circumsphere ofare concentric, and the sum of their radii is .
Proof Let be a spherical body of constant width, and be its circumsphere, where . Since the diameter of is , we have . Let be the ball concentric with having radius . By Lemma 5, is contained in . We will prove that is an insphere of and is unique. Suppose that it is not; then there is a ball different from , with radius . Then, by Lemma 4, there exists a sphere concentric with and having radius , which is a contradiction, since and is different from .
Remark 1 Bodies of constant width in an dimensional Riemannian manifold were introduced and studied by Dekster^{[12]}. Then Dekster studied the incenter and circumcenter of bodies of constant width in . He proved that each circumcenter ofis an incenter and vice versa, and the inradius and circumradius fulfill.
The following is the theorem of Jung's type for spherical space.
Theorem 2 ^{[13]} Letbe a compact set inof diameterand circumradius . Let be a ball of radius containing . Then
(ii)if and only if there exist points on the boundary ofsuch that these points are of equidistant . In other words, contains andimensional spherical regular simplex with diameter .
Remark 2 The definition of body of constant width in Theorem 2 is similar to the definition of body of constant diameter in Section 2. By Lemma 3, we know that the two notions coincide.
4 The Minkowski Measure of Spherical Body of Constant Width
In this section, we firstly give a definition of the Minkowski measure of asymmetry of spherical body of constant width, then we prove the main Theorem given in Section 0, that is Theorem 3 in this section.
Definition 1 Denote bybe an sconvex body of constant width . Letbe the radius of the circumsphere of. The Minkowski measure of asymmetry ofis defined by
Remark 3
(i) We have not found a suitable definition of the Minkowski measure of asymmetry for general spherical convex bodies;
(ii) The Definition 1 is motivated by the work of Brandenberg and Merino^{[2]}.
In the following, we prove our main result.
Theorem 3 Let be an sconvex body of constant width. Then,
The equality holds on the lefthand side if and only if is a spherical ball, and on the righthand side if and only if is a completion of a spherical regular simplex.
Proof Let be the constant width of , and , the radii of the insphere, circumsphere respectively. Then , which implies that . We have if and only if . Hence, if and only if is a spherical ball.
By Theorem 2, we have , then
By Theorem 2, the equality holds in the above inequality if and only if contains a spherical regular simplex of diameter , which implies that is a completion of a spherical regular simplex.
Corollary 1 Letbe an sconvex body of constant width. Then,
The equality holds on the lefthand side if and only ifis a spherical disc, and on the righthand side if and only ifis a spherical Reuleaux triangle.
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