Constantin Zălinescu
In this short note we provide a new proof of the recent result of Han and Nashimura on the separation of spherical convex sets established in arXiv:2002.06558. Our proof is based on a result stated in locally convex spaces.
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A characterization of convex hyperbolic polyhedra and of convex polyhedra inscribed in the sphere
我们描述了一个表征凸多面体在$\h^3$在他们的二面角，Rivin开发。我们还描述了该理论的一些几何和组合的后果。这些后果之一是一个组合特征的凸多面体在$\E^3$所有的顶点位于单位球。这解决了雅各布·施泰纳在1832年提出的一个问题。
An interval version of separation by semispaces in max-min convexity
利用极大极小半空间研究了极大极小凸集中闭盒的分离问题。这可以看作是已知分离结果的区间推广。在盒和极大极小凸集满足一定条件的情况下，给出了分离性的构造性证明，并证明了当这个条件不成立时，分离性是不可能的.我们还研究了极大极小凸集的盒分离和盒与半空间的分离。
Perimeter approximation of convex discs in the hyperbolic plane and on the sphere
研究双曲平面和球面上凸盘的周长近似问题，证明双曲平面中凸盘的最佳近似多边形始终内接于盘，而球面则不成立。
A Lower Bound on the Waist of Unit Spheres of Uniformly Convex Normed Spaces
本文利用余维数大于1的局部化技巧和Borsuk-Ulam定理的一个强形式，给出了一致凸赋范空间单位球面束腰的一个下界.本文使用的工具遵循M. Gromov在[4]。我们的等周型不等式推广了[5]中的Gromov-Milman等周不等式.
We generalize Cauchy's celebrated theorem on the global rigidity of convex polyhedra in Euclidean $3$-space $\mathbb{E}^{3}$ to the context of circle polyhedra in the $2$-sphere $\mathbb{S}^{2}$. We prove that any two convex and proper non-unitary c-polyhedra with Möbius-congruent faces that are consistently oriented are Möbius-congruent. Our result implies the global rigidity of convex inversive distance circle packings in the Riemann sphere as well as that of certain hyperideal hyperbolic polyhedra in $\mathbb{H}^{3}$.
Stability of Convex Spheres
证明所有简单闭测地线长度接近2π的严格凸2-球面在C⁰Cheeger-Gromov意义下接近标准球面，揭示了凸球面的稳定性性质。
We use some recent work of Fung and an earlier paper of Hadwiger, Larman and Mani to extend the class of $n$-spheres which are known to inscribe an $(n + 1)$-rhomb in a given direction.
The topic of totally separable sphere packings is surveyed with a focus on regular constructions, uniform tilings, and contact number problems. An enumeration of all regular totally separable sphere packings in $\mathbb{R}^2$, $\mathbb{R}^3$, and $\mathbb{R}^4$ which are based on convex uniform tessellations, honeycombs, and tetracombs, respectively, is presented, as well as a construction of a family of regular totally separable sphere packings in $\mathbb{R}^d$ that is not based on a convex uniform $d$-honeycomb for $d \geq 3$.
On hyperplanes and semispaces in max-min convex geometry
超平面分离的概念是凸几何及其热带（最大）类似物的基础。然而，类似的分离结果在最大-最小凸几何是基于半空间。本文回答了极大极小凸几何中哪些半空间是超平面以及何时可以用超平面经典分离的问题。
Cyclic projectors and separation theorems in idempotent convex geometry
幂等元半环上的半模，如极大加半环或热带半环，与凸锥有许多共同之处。这种类比在极大加半环的n重笛卡儿积的子半模的情况下尤其明显。我们知道可以从不包含向量的闭子半模中分离向量。我们在这里建立了一个更一般的分离定理，它适用于任何具有平凡交的闭半模的有限集合。为了证明这个定理，我们研究了某些非线性算子的谱性质，这里称为幂等循环投影算子。这些是凸分析中已知的循环最近点投影的幂等类似物。幂等循环投影算子的谱可用Hilbert射影度量的适当推广来刻画.我们推导出作为推论我们的主要结果的幂等类似的Helly定理。
On packing spheres into containers (about Kepler's finite sphere packing problem)
在欧氏$d$ -空间中，容器问题要求将大小相等的$n$球装入一个最小扩张的固定容器中。如果容器是一个光滑的凸体和$d\geq 2$，我们表明，解决方案的容器问题不能有一个"简单的结构“为大$n$。通过这一点，我们特别发现，存在任意小的$r>0$，使得在具有最大数量的半径为$r$的球的光滑三维凸体中的填充不一定是六方密填充。这与开普勒的著名声明相矛盾，即立方体或六边形紧密堆积“将是最紧密的可能，因此在没有其他安排的情况下，更多的球体可以装入同一个容器”。
Let $\mathcal{P}$ and $\mathcal{P}'$ be $3$-dimensional convex polytopes in $\mathbb{R}^3$ and $S \subseteq \mathbb{R}^3$ be a non-empty intersection of an open set with a sphere. As a consequence of a somewhat more general result it is proved that $\mathcal{P}$ and $\mathcal{P}'$ coincide up to translation and/or reflection in a point if $|\int_{\mathcal{P}} e^{-i\mathbf{s}\cdot\mathbf{x}} \,\mathbf{dx}| = |\int_{\mathcal{P}'} e^{-i\mathbf{s}\cdot\mathbf{x}} \,\mathbf{dx}|$ for all $\mathbf{s} \in S$. This can be applied to the field of crystallography regarding the question whether a nanopar…
In this paper, we show that the $C^1$-differentiability of the norm of a two-dimensional normed space depends only on distances between points of the unit sphere in two different ways. As a consequence, we see that any isometry between the spheres of normed planes $τ:S_X\to S_Y$ is linear, provided that there exist linearly independent $x,\overline{x}\in S_X$ where $S_X$ is not differentiable and that $S_X$ is piecewise differentiable. We end this work by showing that the isometry $τ:C_X\to C_Y$ is linear even if it is not an isometry between spheres: every isometry between (planar) Jordan pie…
The goal of this paper is to present a lower bound for the Mahler volume of at least 4-dimensional symmetric convex bodies. We define a computable dimension dependent constant through a 2-dimensional variational (max-min) procedure and demonstrate that the Mahler volume of every (at least 4-dimensional) symmetric convex body is greater than a (simple) function of this constant. Similar to the proof of Gromov's Waist of the Sphere Theorem in [18], our result is proved via localisation-type arguments obtained from a suitable measurable partition (or partitions) of the canonical sphere.
We say that a tiling separates discs of a packing in the Euclidean plane, if each tile contains exactly one member of the packing. It is a known elementary geometric problem to show that for each locally finite packing of circular discs, there exists a separating tiling with convex polygons. In this paper we show that this separating property remains true for circle packings on the sphere and in the hyperbolic plane. Moreover, we show that in the Euclidean plane circles are the only convex discs, whose packings with similar copies can be always separated by polygonal tilings. The analogous sta…
Shortest closed curve to contain a sphere in its convex hull
本文证明了欧几里得3维空间中包含单位球体于其凸包内的闭曲线长度下界为$4π$，推广了Zalgaller猜想的解答，并给出了$n$维情形下的精确估计。
Surface areas of equifacetal polytopes inscribed in the unit sphere $\mathbb{S}^2$
研究单位球面上7、8个点的凸包表面积最大化问题，给出了凸包为全等等腰或等边三角形面时的最优解。
Given $n$ distinct points $\mathbf{x}_1, \ldots, \mathbf{x}_n$ in $\mathbb{R}^d$, let $K$ denote their convex hull, which we assume to be $d$-dimensional, and $B = \partial K $ its $(d-1)$-dimensional boundary. We construct an explicit one-parameter family of continuous maps $\mathbf{f}_{\varepsilon} \colon \mathbb{S}^{d-1} \to K$ which, for $\varepsilon > 0$, are defined on the $(d-1)$-dimensional sphere and have the property that the images $\mathbf{f}_{\varepsilon}(\mathbb{S}^{d-1})$ are codimension $1$ submanifolds contained in the interior of $K$. Moreover, as the parameter $\varepsilon$ …
We prove an explicit combinatorial formula for the expected number of faces of the zero polytope of the homogeneous and isotropic Poisson hyperplane tessellation in $\mathbb R^d$. The expected $f$-vector is expressed through the coefficients of the polynomial $$ (1+ (d-1)^2x^2) (1+(d-3)^2 x^2) (1+(d-5)^2 x^2) \ldots. $$ Also, we compute explicitly the expected $f$-vector and the expected volume of the spherical convex hull of $n$ random points sampled uniformly and independently from the $d$-dimensional half-sphere. In the case when $n=d+2$, we compute the probability that this spherical conve…
Let $U_1,U_2,\ldots$ be random points sampled uniformly and independently from the $d$-dimensional upper half-sphere. We show that, as $n\to\infty$, the $f$-vector of the $(d+1)$-dimensional convex cone $C_n$ generated by $U_1,\ldots,U_n$ weakly converges to a certain limiting random vector, without any normalization. We also show convergence of all moments of the $f$-vector of $C_n$ and identify the limiting constants for the expectations. We prove that the expected Grassmann angles of $C_n$ can be expressed through the expected $f$-vector. This yields convergence of expected Grassmann angles…
On the shadow boundary of a centrally symmetric convex body
讨论了中心对称凸球$K$（实际上是Minkowski赋范空间的单位球）关于欧氏n-空间$R^n$的方向${\bf x}$的阴影边界的概念.我们引入了对应于这个方向的$K$的广义参数球面的概念，并证明了如果所有非退化的广义参数球面都是拓扑流形，则阴影边界也是拓扑流形.在这种情况下，利用胞状映射的逼近定理，我们得到它们同胚于$(n-2)$维球面$S^{(n-2)}$。我们还证明了在${\bf x}$方向上的平分线（对应赋范空间的等距集）同胚于$R^{(n-1)}$当且仅当所有非退化的一般参数球面都是$(n-2)$ -流形，这意味着如果平分线是$R^{(n-1)}$的同胚复制，则对应的阴影边界是拓扑$(n-2)$ -球面.
Face numbers of 4-Polytopes and 3-Spheres
本文讨论了四维凸多胞体和胞腔3-球面的f-向量和flag-向量。我们提出了两个重要的胖度和复杂度参数：当边和2-面的个数比顶点和小平面的个数多时，胖度F（P）：=（f_1+f_2-20）/（f_0+f_3-10）大;当每个小平面都有许多顶点，并且每个顶点都在许多小平面中时，复杂度C（P）：=（f_{03} -20）/（f_0+f_3-10）大。最近的研究结果表明，这些参数可能允许一个区分锥的f-或旗向量的-连接欧拉格的长度为5（组合对象），-强经常CW 3球（拓扑对象），-凸4-多面体（离散几何对象），和-理性凸4-多面体（其研究涉及算术方面）。进一步的进展将取决于凸4-多面体的更严格的f-向量不等式的推导。另一方面，我们将需要新的构造方法来产生有趣的多面体，这些多面体远非单纯或简单-例如，非常“胖”或“复杂”的4-多面体。在这个方向上，我将报告有关的建设（从联合工作与迈克尔Joswig，大卫Eppstein和格雷格Kuperberg），产量-强正规CW 3-球的任意大的脂肪，-凸4-多面体的脂肪大于5.048，和-合理凸4-多面体的脂肪大于5- 048。
We improve the estimates for the Ekeland--Hofer--Zehnder capacity of convex bodies by Gluskin and Ostrover. In the course of our argument we show that a closed characteristic of minimal action on the boundary of a centrally symmetric convex body in $\mathbb R^{2n}$ must itself be centrally symmetric, with generalizations to some other types of symmetry.
Gale transform of a starshaped sphere
Gale变换是凸几何中一个简单而强大的工具。特别地，Gale变换的使用是在具有较少顶点的多面体的分类中的主要论点。Gale变换的定义及其与凸多胞体有关的主要性质在许多书籍和文献中都有介绍。但似乎还没有文献研究更一般的物体的Gale变换，如球面的三角剖分。在本文中，我们研究了一类称为星形球的球的Gale变换的性质。
Upper bounds for packings of spheres of several radii
给出了单位球面中不同球冠的填充密度和欧几里得空间中不同凸体的平移密度的上界定理。这些定理通过半定规划的应用，推广了球冠和凸体填充的线性规划界。我们进行明确的计算，获得新的边界包装的两种不同大小的球冠和二进制球包装。我们还稍微提高了包装相同的领域的经典问题的界限。
We continue the exploration of the relationship between conformal dimension and the separation profile by computing the separation of families of spheres in hyperbolic graphs whose boundaries are standard Sierpiński carpets and Menger sponges. In all cases, we show that the separation of these spheres is $n^{\frac{d-1}{d}}$ for some $d$ which is strictly smaller than the conformal dimension, in contrast to the case of rank 1 symmetric spaces of dimension $\geq 3$. The value of $d$ obtained naturally corresponds to a previously known lower bound on the conformal dimension of the associated frac…
A closed quasigeodesic on a convex polyhedron is a closed curve that is locally straight outside of the vertices, where it forms an angle at most $π$ on both sides. While the existence of a simple closed quasigeodesic on a convex polyhedron has been proved by Pogorelov in 1949, finding a polynomial-time algorithm to compute such a simple closed quasigeodesic has been repeatedly posed as an open problem. Our first contribution is to propose an extended definition of quasigeodesics in the intrinsic setting of (not necessarily convex) polyhedral spheres, and to prove the existence of a weakly sim…
Separation axiom $S_3$ for geodesic convexity in graphs
本文研究图测地凸性中的 $S_3$ 分离公理，刻画 $S_3$-图及其半空间结构，并针对满足三角条件和网格图给出结构判定、枚举方法及NP完全分离问题的解法。
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