Sami Mezal Almohammad, Márton Naszódi, Zsolt Lángi
Steinitz's theorem states that a graph $G$ is the edge-graph of a $3$-dimensional convex polyhedron if and only if, $G$ is simple, plane and $3$-connected. We prove an analogue of this theorem for ball polyhedra, that is, for intersections of finitely many unit balls in $\mathbb{R}^3$.
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Quantitative Steinitz theorem: A spherical version
本文将施泰尼茨定理的定量欧氏版本推广到球面情形，证明欧氏空间中凸包包含原点且球半径有下界的结果可推出对应的球面版本。
Quantitative Steinitz Theorem: A polynomial bound
针对 Bárány、Katchalski 和 Pach 提出的定量 Steinitz 定理猜想，本文证明至多选取 $2d$ 个顶点所得凸包包含半径为 $1/(5d^2)$ 的球，给出首个多项式下界，并给出 $2/\sqrt d$ 的上界。
We study two notions. One is that of spindle convexity. A set of circumradius not greater than one is spindle convex if, for any pair of its points, it contains every short circular arc of radius at least one, connecting them. The other objects of study are bodies obtained as intersections of finitely many balls of the same radius, called ball-polyhedra. We find analogues of several results on convex polyhedral sets for ball-polyhedra.
The rigidity theorems of Alexandrov (1950) and Stoker (1968) are classical results in the theory of convex polyhedra. In this paper we prove analogues of them for normal (resp., standard) ball-polyhedra. Here, a ball-polyhedron means an intersection of finitely many congruent balls in Euclidean 3-space.
A volume formula for Reuleaux polyhedra
本文推导了Reuleaux多面体的体积公式，基于其边长计算其体积，拓展了球多面体体积研究的范围。
A ball-polyhedron is the intersection with non-empty interior of finitely many (closed) unit balls in Euclidean 3-space. One can represent the boundary of a ball-polyhedron as the union of vertices, edges, and faces defined in a rather natural way. A ball-polyhedron is called a simple ball-polyhedron if at every vertex exactly three edges meet. Moreover, a ball-polyhedron is called a standard ball-polyhedron if its vertex-edge-face structure is a lattice (with respect to containment). To each edge of a ball-polyhedron one can assign an inner dihedral angle and say that the given ball-polyhedro…
On basic $r$-ball polyhedra
本文研究了$d$维空间中基本$r$-球体多面体的面结构，确定了最大$ i $维面数量，并证明了对于$d>2$，此类多面体具有全局刚性。
A very fundamental geometric problem on finite systems of spheres was independently phrased by Kneser (1955) and Poulsen (1954). According to their well-known conjecture if a finite set of balls in Euclidean space is repositioned so that the distance between the centers of every pair of balls is decreased, then the volume of the union (resp., intersection) of the balls is decreased (resp., increased). In the first half of this paper we survey the state of the art of the Kneser-Poulsen conjecture in Euclidean, spherical as well as hyperbolic spaces with the emphases being on the Euclidean case.…
Quantitative Steinitz theorem and polarity
本文研究定量 Steinitz 定理中半径 r 的界，利用极性技巧证明当凸多面体顶点数随维数线性增长时，可选取 2d 个顶点使 r 保持多项式下界。
For a Riemannian polyhedra, we study the geometry of the unit ball for the unidimensional stable norm (stable ball). In the case of a unidimensional Riemannian polyhedra (graph), we show that the stable ball is a polytope whose vertices are completely described by combinatorial properties of the graph. We study then the realizable forms as stable ball of Riemannan manifolds of dimension larger than three. For a Riemannian manifold $(M, g)$ fixed, we show that a broad class of polytopes can appear as stable ball of metrics in the conformal class of $g$. We use for that a polyhedral technique.
The perimeter and volume of a Reuleaux polyhedron
本文研究球心含最大数量直径对的三维球多面体，用Gauss-Bonnet定理与积分公式计算其周长和体积，并将方法调整用于近似由球多面体构造的Meissner常宽多面体的体积。
We prove a randomized version of the generalized Urysohn inequality relating mean-width to the other intrinsic volumes. To do this, we introduce a stochastic approximation procedure that sees each convex body K as the limit of intersections of Euclidean balls of large radii and centered at randomly chosen points. The proof depends on a new isoperimetric inequality for the intrinsic volumes of such intersections. If the centers are i.i.d. and sampled according to a bounded continuous distribution, then the extremizing measure is uniform on a Euclidean ball. If one additionally assumes that the …
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。
This paper is a study of the interaction between the combinatorics of boundaries of convex polytopes in arbitrary dimension and their metric geometry. Let S be the boundary of a convex polytope of dimension d+1, or more generally let S be a `convex polyhedral pseudomanifold'. We prove that S has a polyhedral nonoverlapping unfolding into R^d, so the metric space S is obtained from a closed (usually nonconvex) polyhedral ball in R^d by identifying pairs of boundary faces isometrically. Our existence proof exploits geodesic flow away from a source point v in S, which is the exponential map to S …
Uniquely realisable graphs in polyhedral normed spaces
本文研究多面体赋范空间中框架的全局刚性问题，提出了判定算法、通用框架的必要条件，并在$\ell_\infty^d$空间中获得了更强的充分条件和二维精确刻画。
In this paper we consider the ball and horoball packings belonging to $3$-dimensional Coxeter tilings that are derived by simply truncated orthoschemes with parallel faces. The goal of this paper to determine the optimal ball and horoball packing arrangements and their densities for all above Coxeter tilings in hyperbolic 3-space $\mathbb{H}^3$. The centers of horoballs are required to lie at ideal vertices of the polyhedral cells constituting the tiling, and we allow horoballs of different types at the various vertices. We prove that the densest packing of the above cases is realized by horob…
The convex hull generated by the restriction to the unit ball of a stationary Poisson point process in the $d$-dimensional Euclidean space is considered. By establishing sharp bounds on cumulants, exponential estimates for large deviation probabilities are derived and the relative error in the central limit theorem on a logarithmic scale is investigated for a large class of key geometric characteristics. This includes the number of lower-dimensional faces and the intrinsic volumes of the random polytopes. Furthermore, moderate deviation principles for the spatial empirical measures induced by …
We study metric properties of convex bodies B and their polars B^o, where B is the convex hull of an orbit under the action of a compact group G. Examples include the Traveling Salesman Polytope in polyhedral combinatorics (G=S_n, the symmetric group), the set of non-negative polynomials in real algebraic geometry (G=SO(n), the special orthogonal group), and the convex hull of the Grassmannian and the unit comass ball in the theory of calibrated geometries (G=SO(n), but with a different action). We compute the radius of the largest ball contained in the symmetric Traveling Salesman Polytope, g…
Selected topics from the theory of intersections of balls
本文讨论了球体交集理论中的Kneser-Poulsen猜想、Blaschke-Santaló型不等式以及球多面体的体积性质。
In this paper we solve several reverse isoperimetric problems in the class of $λ$-convex bodies, i.e., convex bodies whose curvature at each point of their boundary is bounded below by some $λ> 0$. We give an affirmative answer in $\mathbb{R}^3$ to a conjecture due to Borisenko which states that the $λ$-convex lens, i.e., the intersection of two balls of radius $1/λ$, is the unique minimizer of volume among all $λ$-convex bodies of given surface area. Also, we prove a reverse inradius inequality: in model spaces of constant curvature and arbitrary dimension, we show that the $λ$-convex lens (p…
Optimal Lagrange Interpolation Projectors and Legendre Polynomials
本文研究了凸体上最优Lagrange插值投影子范数的上下界，利用Legendre多项式的几何刻画证明了任意凸体的显式下界，并给出了球和立方体情形的精确或渐近估计。
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