Ralph Howard
In 1926 S. Nakajima (= A. Matsumura) showed that any convex body in $\R^3$ with constant width, constant brightness, and boundary of class $C^2$ is a ball. We show that the regularity assumption on the boundary is unnecessary, so that balls are the only convex bodies of constant width and brightness.
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For a convex body $K\subset\R^n$, the $k$th projection function of $K$ assigns to any $k$-dimensional linear subspace of $\R^n$ the $k$-volume of the orthogonal projection of $K$ to that subspace. Let $K$ and $K_0$ be convex bodies in $\R^n$, and let $K_0$ be centrally symmetric and satisfy a weak regularity and curvature condition (which includes all $K_0$ with $\f K_0$ of class $C^2$ with positive radii of curvature). Assume that $K$ and $K_0$ have proportional 1st projection functions (i.e., width functions) and proportional $k$th projection functions. For $2\le k<(n+1)/2$ and for $k=3, n=5…
The X-ray numbers of some classes of convex bodies are investigated. In particular, we give a proof of the X-ray Conjecture as well as of the Illumination Conjecture for almost smooth convex bodies of any dimension and for convex bodies of constant width of dimensions 3, 4, 5 and 6.
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.
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 …
Hyperbolic width functions and characterizations of bodies of constant width in the hyperbolic space
本文研究n维双曲空间中的多种宽度函数，分析其连续性等性质，提出推广Leichtweiss宽度函数的新双曲宽度，并据此证明常宽体的刻画定理。
Following Santaló's approach, we prove several characterizations of a disc among bodies of constant width, constant projections lengths, or constant section lengths on given families of geodesics.
On extreme constant width bodies in $\mathbb{R}^3$
本文研究了$\mathbb{R}^3$中在Minkowski加法下的极值常宽体，证明了Meissner多面体及对称Reuleaux多边形旋转体均为极值，并猜想了一般刻画。
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.
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.
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…
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 $π/{2}$.
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.
Makeev conjectured that every constant-width body is inscribed in the dual difference body of a regular simplex. We prove that homologically, there are an odd number of such circumscribing bodies in dimension 3, and therefore geometrically there is at least one. We show that the homological answer is zero in higher dimensions, a result which is inconclusive for the geometric question. We also give a partial generalization involving affine circumscription of strictly convex bodies.
Lipschitzness of the width and diameter functions of convex bodies in $\mathbb R^n$
研究了凸体在R^n中宽度和直径函数的Lipschitz性，通过直径和厚度确定其Lipschitz常数，并提出厚度的对偶方法。
Strengthened inequalities for the mean width and the $\ell$-norm of origin symmetric convex bodies
本文改进了对称凸体均宽和ℓ-范数的极值结果，证明了立方体和正交叉多面体在特定条件下的最优稳定性版本，并探讨了相关稳定性结果。
A new family of constant width bodies obtained from Reuleaux polyhedra
该研究提出一种从单位勒洛多面体构造常宽体的新方法，其保留原多面体对称性，无需选择对偶边，对勒洛四面体可得到特定常宽体。
We present an alternative proof of the following fact: the hyperspace of compact closed subsets of constant width in $\mathbb R^n$ is a contractible Hilbert cube manifold. The proof also works for certain subspaces of compact convex sets of constant width as well as for the pairs of compact convex sets of constant relative width. Besides, it is proved that the projection map of compact closed subsets of constant width is not 0-soft in the sense of Shchepin, in particular, is not open.
In this paper we state a one-to-one connection between the maximal ratio of the circumradius and the diameter of a body (the Jung constant) in an arbitrary Minkowski space and the maximal Minkowski asymmetry of the complete bodies within that space. This allows to generalize and unify recent results on complete bodies and to derive a necessary condition on the unit ball of the space, assuming a given body to be complete. Finally, we state several corollaries, i.e. concerning the Helly dimension or the Banach-Mazur distance.
Small Volume Bodies of Constant Width with Tetrahedral Symmetries
本文构造了具有四面体对称性的常宽小体积体，其中U_3的体积比其他类似体更小，且比梅斯纳体稍大。
It is known that an $n$-dimensional convex body which is typical in the sense of Baire category, shows a simple, but highly non-intuitive curvature behaviour: at almost all of its boundary points, in the sense of measure, all curvatures are zero, but there is also a dense and uncountable set of boundary points at which all curvatures are infinite. The purpose of this paper is to find a counterpart to this phenomenon for typical convex bodies of given constant width. Such bodies cannot have zero curvatures. A main result says that for a typical $n$-dimensional convex body of constant width $1$ …
Area Minimization Among Group-Invariant Planar Convex Bodies of Constant Width
该研究在给定对称约束下，分类了固定原点的有限等距变换群G作用下不变的常宽度凸体中的最小面积者，明确了不同G对应的极小值类型。
Note on illuminating constant width bodies
本研究推广已有构造方法，改进了$\mathbb{R}^n$中常宽体照明数的指数下界，得到至少为$(τ+o(1))^n$（$τ\approx1.047$）的新结果。
Small volume bodies of constant width
该研究针对高维常宽度体体积下界的相关问题，显式构造了体积小于$0.9^n$倍单位球体积的2宽度常宽度体，解决了O. Schramm提出的相关疑问。
Width of convex bodies in hyperbolic space
本文研究双曲空间中凸体的宽度、厚度、常宽体、约化体、完备体与常直径体，证明常宽体严格凸且常宽条件等价于常直径与完备性，并证明双曲正四面体不是约化体。
Let n be a natural number equal or greater than 2. In this paper we study the topological structure of certain hyperspaces of convex subsets of constant width, equipped with the Hausdorff metric topology. We focus our attention on the hyperspace cw_D(R^n) of all compact convex subsets with constant width d\in D, where D is a convex subset of [0,\infty). Our main result states that cw_D(R^n) is homeomorphic to D\times R^n\times Q, where Q denotes the Hilbert cube. We also prove that the hyperspace crw_D(R^n), consisting of all pairs of compact convex sets of constant relative width d\in D is ho…
Convex bodies of constant width with exponential illumination number
本文证明了在$\mathbb{E}^n$中存在照明数至少为$(\cos(\pi/14)+o(1))^{-n}$的定宽凸体，并证明了存在直径为1的有限集无法被$(2/\sqrt{3}+o(1))^n$个直径为1的球覆盖，回答了G. Kalai的问题并改进了已有结果。
On reduced spherical bodies
本论文汇集五项研究，系统探讨 $S^d$ 上约化球面凸体及等宽体的形状、性质、直径与覆盖圆盘半径。
Bounding the diameter-width ratio using containment inequalities of means of convex bodies
本文研究了平面伪完全集的直径-宽度比，通过Minkowski中心凸集的包含不等式，定义了τ(K)并给出了其与闵可夫斯基不对称性相关的取值区域。
Lattice Reduced and Complete Convex Bodies
本文研究格约化与格完备凸体的结构性质，证明实现平坦度常数的单纯形必为格约化，并给出顶点数上界、独立方向数下界及构造方法。
We prove several estimates for the volume, mean width, and the value of the Wills functional of sections of convex bodies in John's position, as well as for their polar bodies. These estimates extend some well-known results for convex bodies in John's position to the case of lower-dimensional sections, which had mainly been studied for the cube and the regular simplex. Some estimates for centrally symmetric convex bodies in minimal surface area position are also obtained.
Pál's isominwidth inequality for ball convex bodies in planes of constant curvature
本文在欧氏、双曲和球面平面中，对r-球凸体证明了等宽不等式，推广了Pál经典结果。
From inequalities relating symmetrizations of convex bodies to the diameter-width ratio for complete and pseudo-complete convex sets
本文研究平面Minkowski中心凸紧集的对称化相关因子α(K)与不对称性的关系，证明不对称性大于黄金分割比时K与-K边界恰交于6点，并推导了完备与伪完备集直径-宽度比的更紧界。
Strengthened inequalities for the mean width and the $\ell$-norm
本文加强了关于凸体均宽和ℓ-范数的不等式，证明了正则单纯形在特定条件下的极值性质，并研究了中心各向同性测度支撑集的凸包的相关稳定性结果。
A four-dimensional body of constant width
本文提出了一种在R^4中构造常宽体的新几何方法，通过修改四维Reuleaux单纯形得到具有四面体对称性的第二Meissner体的自然类比，并分析了其三维投影的性质。
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 $\fracπ{2}$. …
Spherical coverings and X-raying convex bodies of constant width
本文通过构造球面上少于 $2^n$ 个球冠的原点对称覆盖，结合 Schramm 的照明数上界，解决了 $7\le n\le 15$ 维常宽凸体的 X-射线猜想与照明猜想，并给出计算球面离散点集覆盖半径的高效方法。
An in-depth study of ball-bodies
本文深入研究n维欧氏空间中的球形体类，探讨其对偶算子、相关几何不等式、边界结构、对称化方法，并推进了常宽体相关问题的研究。
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.
Denote by ${\mathcal K}^d$ the family of convex bodies in $E^d$ and by $w(C)$ the minimal width of $C \in {\mathcal K}^d$. We ask for the greatest number $Λ_n ({\mathcal K}^d)$ such that every $C \in {\mathcal K}^d$ contains a polytope $P$ with at most $n$ vertices for which $Λ_n ({\mathcal K}^d) \leq \frac{w(P)}{w(C)}$. We give a lower estimate of $Λ_n ({\mathcal K}^d)$ for $n \geq 2d$ based on estimates of the smallest radius of $\big\lfloor {\frac{n}{2}} \big\rfloor$ antipodal pairs of spherical caps that cover the unit sphere of $E^d$. We show that $Λ_3 ({\mathcal K}^2) \geq {\frac 1 2}(3-…
The density of Meissner polyhedra
本文研究三维 Meissner 多面体，证明其在 Hausdorff 拓扑下于常宽体空间中稠密，并以现代视角重新审视 Sallee 的结果，结合球多面体与常宽体构造的最新进展。
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 $δ$.
Approximation of spherical convex bodies of constant width $π/2$
本文针对常宽π/2的球面凸体，证明其可由常宽π/2的球面多面体在任意Hausdorff距离精度下逼近。
The purpose of this paper is to describe a new $3$-dimensional family of bodies of constant width that we have called peabodies, obtained from the Reuleaux tetrahedron by replacing a small neighborhood of all six edges with sections of an envelope of spheres. This family contains, in particular, the two Meissner solids and a body with tetrahedral symmetry that we have called Robert's body. Behind the construction of this family lies the classical notion of confocal quadrics discussed, for example, by Hilbert in his famous book. We study confocal quadrics and prove that the distances of an alte…
On a formula for all sets of constant width in 3d
本文基于二维常宽集合的构造公式，提出依赖 S^2 上函数的三维常宽体边界显式参数化方法，并借助“阴影域”工具证明其完备性。
The "old-new" concept of convex-hull function was investigated by several authors in the last seventy years. A recent research on it led to some other volume functions as the covariogram function, the widthness function or the so-called brightness functions, respectively. A very interesting fact that there are many long-standing open problems connected with these functions whose serious investigation closed before the "age of computers". In this survey, we concentrate only on the three-dimensional case, we will mention the most important concepts, statements, and problems.
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.
Random Width and Brightness: Polyhedral Density Theory, Reconstruction, and Gaussian Identifiability
本文为三维凸体的随机宽度与亮度建立正逆向理论，推导多面体精确密度，并证明基于低阶矩的逆向重构与高斯相关矩阵可辨识性定理。
Optimal mean width and metric entropy estimates for convex bodies
该研究针对凸体，证明了球面平均宽度与体积半径比值的最优界，以及欧氏覆盖数对数的最大值由单纯形和交叉多面体取得，证明采用了 Eldan 的随机局部化方法。
Exact Flatness Constant for One-Point Convex Bodies and the Discrete Isominwidth Problem: The Planar Case
研究整数规划中凸体最多含k个内格点时的最大格宽，证明平面凸体含至多一个内点时格宽不超过3，并导出等最小宽度不等式。
Area bounds for planar convex bodies containing a fixed number of interior integral points
本文研究包含固定数量内部整点的平面凸体的面积界问题，利用格宽数据证明面积界，并给出有理多边形依赖于分母的精确面积界及基于诺特公式的格多边形面积方程。
Floating bodies and duality in spaces of constant curvature
该研究在d维球面、双曲、de Sitter等常曲率空间中构造p-仿射表面积的自然类似物，证明其为对偶共轭非欧凸体浮动体的体积导数，建立了由常曲率控制的实解析统一框架，揭示了其与欧氏空间两类曲率测度的关联。
In this paper, we establish mean width inequalities of sections and projections of convex bodies for isotropic measures with complete equality conditions, which extends the recent work of Alonso-Gutiérrez and Brazitikos. Different from their approach, our proof is based on the approach developed by Lutwak, Yang and Zhang, by using the Ball-Barthe inequality, the mass transportation, and the isotropic embedding.
The mean width is a measure on n-dimensional convex bodies. An integral formula for the mean width of a regular n-simplex appeared in the electrical engineering literature in 1997. As a consequence, expressions for the expected range of a sample of n+1 normally distributed variables, for n<=6, carry over to widths of regular n-simplices. As another consequence, precise asymptotics for the mean width become available as n->infty.
The notion of ball convexity, considered in finite dimensional real Banach spaces, is a natural and useful extension of usual convexity; one replaces intersections of half-spaces by suitable intersections of balls. A subset $S$ of a normed space is called ball convex if it coincides with its ball hull, which is obtained as intersection of all balls (of fixed radius) containing $S$. Ball convex sets are closely related to notions like ball polytopes, complete sets, bodies of constant width, and spindle convexity. We will study geometric properties of ball convex bodies in normed spaces, for exa…
Upper Bounds on Covering Minima of Convex Bodies
本文给出了凸体覆盖极值的两个新上界，基于投影和线性子空间的覆盖极值。结果适用于两个凸体的直接和，推广了之前关于直接和的覆盖半径和格宽的研究，并应用于标准终端单纯形，缩小了Gonzaléz Merino和Schymura(2017)猜想中的上下界差距。
We aim to study the classical Rosenthal-Szasz inequality for a plane whose geometry is given by a norm. This inequality states that the bodies of constant width have the largest perimeter among all planar convex bodies of given diameter. In the case where the unit circle of the norm is given by a Radon curve, we obtain an inequality which is completely analogous to the Euclidean case. For arbitrary norms we obtain an upper bound for the perimeter calculated in the anti-norm, yielding an analogous characterization of all curves of constant width. To derive these results, we use methods from the…
A natural model for the approximation of a convex body $K$ in $\mathbb{R}^d$ by random polytopes is obtained as follows. Take a stationary Poisson hyperplane process in the space, and consider the random polytope $Z_K$ defined as the intersection of all closed halfspaces containing $K$ that are bounded by hyperplanes of the process not intersecting $K$. If $f$ is a functional on convex bodies, then for increasing intensities of the process, the expectation of the difference $f(Z_K)-f(K)$ may or may not converge to zero. If it does, then the order of convergence and possible limit relations are…
The mean width is a measure on three-dimensional convex bodies that enjoys equal status with volume and surface area [Rota]. As the phrase suggests, it is the mean of a probability density f. We verify formulas for mean widths of the regular tetrahedron and the cube. Higher-order moments of f_tetra and f_cube have not been examined until now. Assume that each polyhedron has edges of unit length. We deduce that the mean square width of the regular tetrahedron is 1/3+(3+sqrt(3))/(3*pi) and the mean square width of the cube is 1+4/pi.
A new intrinsic volume metric is introduced for the class of convex bodies in $\mathbb{R}^n$. As an application, an inequality is proved for the asymptotic best approximation of the Euclidean unit ball by arbitrarily positioned polytopes with a restricted number of vertices under this metric. This result improves the best known estimate, and shows that dropping the restriction that the polytope is contained in the ball or vice versa improves the estimate by at least a factor of dimension. The same phenomenon has already been observed in the special cases of volume, surface area and mean width …
A solution to Bezdek's conjecture
本文证明在n≥2时，给定内径的λ-凸体中λ-凸透镜平均宽度最大，肯定回答Bezdek猜想；附加对称性时解决其任意内蕴体积的类似猜想；还证明固定外接半径的λ-凸体中λ-凸纺锤体平均宽度唯一最小。
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