相关论文: The density of Meissner polyhedra
We describe a characterization of convex polyhedra in $\h^3$ in terms of their dihedral angles, developed by Rivin. We also describe some geometric and combinatorial consequences of that theory. One of these consequences is a combinatorial…
The Kneser-Poulsen conjecture says that if a finite collection of balls in a Euclidean (spherical or hyperbolic) space is rearranged so that the distance between each pair of centers does not increase, then the volume of the union of these…
A stationary Boolean model is the union set of random compact particles which are attached to the points of a stationary Poisson point process. For a stationary Boolean model with convex grains we consider a recently developed collection of…
We determine when a convex body in $\mathbb{R}^d$ is the closed unit ball of a reasonable crossnorm on $\mathbb{R}^{d_1}\otimes\cdots\otimes\mathbb{R}^{d_l},$ $d=d_1\cdots d_l.$ We call these convex bodies "tensorial bodies". We prove that,…
Let $C\subset \mathbb{S}^2$ be a spherical convex body of constant width $\tau$. It is known that (i) if $\tau<\pi/2$ then for any $\varepsilon>0$ there exists a spherical convex body $C_\varepsilon$ of constant width $\tau$ whose boundary…
The purpose of this paper is to introduce the notion of mixed twistor structure, a generalization of the notion of mixed Hodge structure. The utility of this notion is to make possible a theory of weights for various things surrounding…
We prove that if a topological sphere smoothly embedded into $\mathbb{R}^3$ with normal curvatures absolutely bounded by $1$ is contained in an open ball of radius $2$, then the region it bounds must contain a unit ball. This result…
In \cite{Sz17-2} we proved that to each saturated congruent hyperball packing exists a decomposition of $3$-dimensional hyperbolic space $\mathbb{H}^3$ into truncated tetrahedra. Therefore, in order to get a density upper bound for…
We develop a Morse-Lusternik-Schnirelmann theory for the distance between two points of a smoothly embedded circle in a complete Riemannian manifold. This theory suggests very naturally a definition of width that generalises the classical…
After work of W. P. Thurston, C. Bavard and \'E. Ghys constructed particular hyperbolic polyhedra from spaces of deformations of Euclidean polygons. We present this construction as a straightforward consequence of the theory of…
A subset $R$ of integers is a set of Bohr recurrence if every rotation on $\mathbb{T}^d$ returns arbitrarily close to zero under some non-zero multiple of $R$. We show that the set $\{k!\, 2^m3^n\colon k,m,n\in \mathbb{N}\}$ is a set of…
In this paper we study the problem of hyperball (hypersphere) packings in $3$-dimensional hyperbolic space. We introduce a new definition of the non-compact saturated ball packings and describe to each saturated hyperball packing, a new…
We characterize the surjective isometries, with respect to the Hausdorff distance, of the class of bodies given by intersections of Euclidean unit balls. We show that any such isometry is given by the composition of a rigid motion with…
A hyperk\"ahler manifold is defined as a Riemannian manifold endowed with three covariantly constant complex structures that are quaternionically related. A twistor space is characterized as a holomorphic fiber bundle $p: \mathcal{Z}…
We study the isoperimetric problem for Euclidean space endowed with a continuous density. In dimension one, we characterize isoperimetric regions for a unimodal density. In higher dimensions, we prove existence results and we derive…
We study symmetrization procedures within the class $\mathcal S_n$ of \emph{ball-bodies}, i.e.\ intersections of unit Euclidean balls (equivalently, summands of the Euclidean unit ball, or $c$-convex sets via the $c$-duality $A\mapsto…
We derive lower estimates for the approximation of the $d$-dimensional Euclidean ball by polytopes with a fixed number of $k$-dimensional faces, $k\in\{0,1,\ldots,d-1\}$. The metrics considered include the intrinsic volume difference and…
In his paper "On the Schlafli differential equality", J. Milnor conjectured that the volume of n-dimensional hyperbolic and spherical simplices, as a function of the dihedral angles, extends continuously to the closure of the space of…
In hyperbolic space density cannot be defined by a limit as we define it in Euclidean space. We describe the local density bounds for sphere packings and we discuss the different attempts to define optimal arrangements in hyperbolic space.
This article is dedicated to prove Buser's conjecture about Bers' constants for spheres with cusps (or marked points) and for hyperelliptic surfaces. More specifically, our main theorem states that any hyperbolic sphere with $n$ cusps has a…