We present a survey article about the geometry of convex bodies on the $d$-dimensional sphere $S^d$. We concentrate on the results based on the notion of the width of a convex body $C \subset S^d$ determined by a supporting hemisphere of $C$. Important tools are the lunes containing $C$. The supporting hemispheres take over the role of the supporting half-spaces of a convex body in Euclidean space, and lunes the role of strips. Also essential is the notion of thickness of $C$, i.e, its minimum width. In particular, we describe properties of reduced spherical convex bodies and spherical bodies of constant width. The last notion coincides with the notions of complete bodies and bodies of constant diameter on $S^d$. The reminded and commented here results concern mostly the width, thickness, diameter, perimeter, area and extreme points of spherical convex bodies, reduced bodies and bodies of constant width.
The aim of this paper is to present some properties of reduced spherical convex bodies on the two-dimensional sphere $S^2$. The intersection of two different non-opposite hemispheres is called a lune. By its thickness we mean the distance of the centers of the two semicircles bounding it. The thickness $Δ(C)$ of $C$ is the minimum thickness of a lune containing $C$. We say that a spherical convex body $R$ is reduced if $Δ(Z) < Δ(R)$ for every spherical convex body $Z \subset R$ different from $R$. Our main theorem permits to describe the shape of reduced bodies of thickness below $\fracπ{2}$. …
We present a spherical version of the theorem of Blaschke that every body of constant width $w < \fracπ{2}$ can be approximated as well as we wish in the sense of the Hausdorff distance by a body of constant width $w$ whose boundary consists only of pieces of circles of radius $w$. This is a special case of our theorem about approximation of spherical reduced bodies.
The spherical centroid body of a centrally-symmetric convex body in the Euclidean unit sphere is introduced. Two alternative definitions - one geometric, the other probabilistic in nature - are given and shown to lead to the same objects. The geometric approach is then used to establish a number of basic properties of spherical centroid bodies, while the probabilistic approach inspires the proof of a spherical analogue of the classical polar Busemann-Petty centroid inequality.
In this paper we show that a spherical convex body $C$ is of constant diameter $τ$ if and only if $C$ is of constant width $τ$, for $0<τ<π$. Moreover, some applications to Wulff shapes are given.
Similarly to the classic notion in $E^d$, a subset of a positive diameter below $\fracπ{2}$ of a hemisphere of the sphere $S^d$ is called complete, provided adding any extra point increases its diameter. Complete sets are convex bodies on $S^d$. Our main theorem says that on $S^d$ complete bodies of diameter $δ$ coincide with bodies of constant width $δ$.
We present some relationships between the diameter, width and thickness of a reduced convex body on the $d$-dimensional sphere. We apply the obtained properties to recognize if a Wulff shape in the Euclidean $d$-space is self-dual.
For any Wulff shape, its dual Wulff shape is naturally defined. A self-dual Wulff shape is a Wulff shape equaling its dual Wulff shape exactly. In this paper, it is shown that a Wulff shape is self-dual if and only if the spherical convex body induced by it is of constant width $π/{2}$.
A subset of the d-dimensional Euclidean space having nonempty interior is called a spindle convex body if it is the intersection of (finitely or infinitely many) congruent d-dimensional closed balls. The spindle convex body is called a "fat" one, if it contains the centers of its generating balls. The core part of this paper is an extension of Schramm's theorem and its proof on illuminating convex bodies of constant width to the family of "fat" spindle convex bodies.
In this note we introduce the problem of illumination of convex bodies in spherical spaces and solve it for a large subfamily of convex bodies. We derive from it a combinatorial version of the classical illumination problem for convex bodies in Euclidean spaces as well as a solution to that for a large subfamily of convex bodies, which in dimension three leads to special Koebe polyhedra.
The intersection $L$ of two different non-opposite hemispheres $G$ and $H$ of a $d$-dimensional 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 %spherical convex body $C \subset S^d$ we define ${\rm width}_G(C)$ as the thickness of the narrowest lune or lunes of the form $G \cap H$ containing $C$. If ${\rm width}_G(C) =w$ 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…
The intersection $L$ of two different non-opposite hemispheres of the unit sphere $S^2$ is called a lune. By $Δ(L)$ we denote the distance of the centers of the semicircles bounding $L$. By the thickness $Δ(C)$ of a convex body $C \subset S^2$ we mean the minimal value of $Δ(L)$ over all lunes $L \supset C$. We call a convex body $R\subset S^2$ reduced provided $Δ(Z) < Δ(R)$ for every convex body $Z$ being a proper subset of $R$. Our aim is to estimate the diameter of $R$, where $Δ(R) < \fracπ{2}$, in terms of its thickness.
This paper contains a new concept to measure the width and thickness of a convex body in the hyperbolic plane. We compare the known concepts with the new one and prove some results on bodies of constant width, constant diameter and given thickness.
If the n-dimensional unit sphere is covered by finitely many spherically convex bodies, then the sum of the inradii of these bodies is at least π. This bound is sharp, and the equality case is characterized.
Let $K\subset\mathbb S^{d-1}$ be a convex spherical body. Denote by $Δ(K)$ the distance between two random points in $K$ and denote by $σ(K)$ the length of a random chord of $K$. We explicitly express the distribution of $Δ(K)$ via the distribution of $σ(K)$. From this we find the density of distribution of $Δ(K)$ when $K$ is a spherical cap.
The present paper aims to solve some problems proposed by Lassak about the reduced spherical polygons. The main result is to show that the regular spherical n-gon has the minimal perimeter among all reduced spherical polygons of fixed thickness less than π/2 and with at most n vertices. In addition, we determine the maximal diameter of every reduced spherical polygons with a fixed thickness less than π/2. We also find the smallest spherical radius that contains every reduced spherical polygons with a fixed thickness less than π/2.
An $r^{\an}$-center of a compact body $\Om$ in an $n$ dimensional Euclidean space is a point that gives an extremal value of the regularized Riesz potential, which is the (Hadamard regularization of) integration on $\Om$ of the distance from the point to the power $\an$. We show that for any real number $\la$ if a compact body is sufficiently close to a ball in the sense of asphericity then the $r^{\an}$-center is unique. We also study the regularized potentials of a unit ball.
We investigate weighted floating bodies of polytopes. We show that the weighted volume depends on the complete flags of the polytope. This connection is obtained by introducing flag simplices, which translate between the metric and combinatorial structure. Our results are applied in spherical and hyperbolic space. This leads to new asymptotic results for polytopes in these spaces. We also provide explicit examples of spherical and hyperbolic convex bodies whose floating bodies behave completely different from any convex body in Euclidean space.
Extending Blaschke and Lebesgue's classical result in the Euclidean plane, it has been recently proved in spherical and the hyperbolic cases, as well, that Reuleaux triangles have the minimal area among convex domains of constant width $D$. We prove an essentially optimal stability version of this statement in each of the three types of surfaces of constant curvature. In addition, we summarize the fundamental properties of convex bodies of constant width in spaces of constant curvature, and provide a characterization in the hyperbolic case in terms of horospheres.
The purpose of this paper is to study the reflections of a convex body. In particular, we are interested in orthogonal reflections of its sections that can be extended to reflections of the whole body. For this reason, we need to study the case of a non-spherical ellipsoid, where a surprising structure arises (Section 2). These results allow us to give a new characterization of ellipsoids in terms of their reflections and, on the other hand, to prove a result deeply related to a conjecture due to K. Bezdek.
We investigate geometrical properties and inequalities satisfied by the complex difference body, in the sense of studying which of the classical ones for the difference body have an analog in the complex framework. Among others we give an equivalent expression for the support function of the complex difference body and prove that, unlike the classical case, the dimension of the complex difference body depends on the position of the body with respect to the complex structure of the vector space. We use spherical harmonics to characterize the bodies for which the complex difference body is a bal…
In this paper, the following two theorems are proved: $(1)$ every spherical convex body $W$ of constant width $Δ(W) \geq \fracπ{2}$ may be covered by a disk of radius $Δ(W) + \arcsin \left( \frac{2\sqrt{3}}{3} \cdot \cos \frac{Δ(W)}{2}\right) - \fracπ{2}$; $(2)$ every reduced spherical convex body $R$ of thickness $Δ(R)<\fracπ{2}$ may be covered by a disk of radius $\arctan \left( \sqrt{2} \cdot \tan \frac{Δ(R)}{2}\right)$.
Dual to Koldobsky's notion of j-intersection bodies, the class of j-projection bodies is introduced, generalizing Minkowski's classical notion of projection bodies of convex bodies. A Fourier analytic characterization of j-projection bodies in terms of their area measures of order j is obtained. In turn, this yields an equivalent characterization of j-projection bodies involving Alesker's Fourier type transform on translation invariant smooth spherical valuations. As applications of these results, several basic properties of j-projection bodies are established and new non-trivial examples are …
We give theorems that can be used to upper bound the densities of packings of different spherical caps in the unit sphere and of translates of different convex bodies in Euclidean space. These theorems extend the linear programming bounds for packings of spherical caps and of convex bodies through the use of semidefinite programming. We perform explicit computations, obtaining new bounds for packings of spherical caps of two different sizes and for binary sphere packings. We also slightly improve bounds for the classical problem of packing identical spheres.
A Minkowski class is a closed subset of the space of convex bodies in Euclidean space Rn which is closed under Minkowski addition and non-negative dilatations. A convex body in Rn is universal if the expansion of its support function in spherical harmonics contains non-zero harmonics of all orders. If K is universal, then a dense class of convex bodies M has the following property. There exist convex bodies T1; T2 such that M + T1 = T2, and T1; T2 belong to the rotation invariant Minkowski class generated by K. It is shown that every convex body K which is not centrally symmetric has a linear …
Rotation intertwining maps from the set of convex bodies in Rn into itself that are continuous linear operators with respect to Minkowski and Blaschke addition are investigated. The main focus is on Blaschke-Minkowski homomorphisms. We show that such maps are represented by a spherical convolution operator. An application of this representation is a complete classification of all even Blaschke-Minkowski homomorphisms which shows that these maps behave in many respects similar to the well known projection body operator. Among further applications is the following result: If an even Blaschke-Min…
The dual Minkowski problem for even data asks what are the necessary and sufficient conditions on an even prescribed measure on the unit sphere for it to be the $q$-th dual curvature measure of an origin-symmetric convex body in $\mathbb{R}^n$. A full solution to this is given when $1 < q < n$. The necessary and sufficient condition is an explicit measure concentration condition. A variational approach is used, where the functional is the sum of a dual quermassintegral and an entropy integral. The proof requires two crucial estimates. The first is an estimate of the entropy integral proved usi…
