相关论文: A new proof of Vazsonyi's conjecture

A new tur\'an-type problem on distances on graphs was introduced by Tyomkyn and Uzzell. In this paper, we focus on the case that the distance is two. We primely show that for any value of $n$, a graph on $n$ vertices without three vertices…

We prove algorithmic and hardness results for the problem of finding the largest set of a fixed diameter in the Euclidean space. In particular, we prove that if $A^*$ is the largest subset of diameter $r$ of $n$ points in the Euclidean…

We prove the convex combination theorem for hyperbolic n-manifolds. Applications are given both in high dimensions and in 3 dimensions. One consequence is that given two geometrically finite subgroups of a discrete group of isometries of…

We prove that every hyperplane passing through the origin in $\rr^{n+1}$ divides an embedded compact free boundary minimal hypersurface of the euclidean $(n+1)$-ball in exactly two connected hypersurfaces. We also show that if a region in…

Let $M^d$ denote the $d$-dimensional Euclidean, hyperbolic, or spherical space. The $r$-dual set of given set in $M^d$ is the intersection of balls of radii $r$ centered at the points of the given set. In this paper we prove that for any…

How can we arrange $n$ lines through the origin in three-dimensional Euclidean space in a way that maximizes the minimum interior angle between pairs of lines? Conway, Hardin and Sloane (1996) produced line packings for $n \leq 55$ that…

A classic theorem of Euclidean geometry asserts that any noncollinear set of $n$ points in the plane determines at least $n$ distinct lines. Chen and Chv\'atal conjectured that this holds for an arbitrary finite metric space, with a certain…

We show that there exist reduced polytopes in three-dimensional Euclidean space. This partially answers the question posed by Lassak on the existence of reduced polytopes in $d$-dimensional Euclidean space for $d\geq 3$. Moreover, we prove…

A well-known conjecture of Caratheodory states that the number of umbilic points on a closed convex surface in ${\mathbb E}^3$ must be greater than one. In this paper we prove this for $C^{3+\alpha}$-smooth surfaces. The Conjecture is first…

We establish a lower bound for the surface area of a closed, convex hypersurface in Euclidean space in terms of its displacement under continuous maps. As a result, a hypothesized lower bound for the volume of a Riemannian $n$-sphere,…

Gr\"unbaum's inequality guarantees that the centroid of a convex body has halfspace depth at least $1/e$: every halfspace containing the centroid captures at least a $1/e$ fraction of the body's volume. For mixed-integer convex sets…

Geometrical objects with integral sides have attracted mathematicians for ages. For example, the problem to prove or to disprove the existence of a perfect box, that is, a rectangular parallelepiped with all edges, face diagonals and space…

The Kneser-Poulsen conjecture says that if a finite collection of balls in a d-dimensional Euclidean space is rearranged so that the distance between each pair of centers does not get smaller, then the volume of the union of these balls…

We prove estimates for the optimal volume of thick embeddings of finite graphs into symmetric spaces, generalising results of Kolmogorov-Barzdin and Gromov-Guth for embeddings into Euclidean spaces. We distinguish two very different…

We solve a problem in the combinatorics of polyhedra motivated by the network simplex method. We show that the Hirsch conjecture holds for the diameter of the graphs of all network-flow polytopes, in particular the diameter of a…

A graph on $2k$ vertices is path-pairable if for any pairing of the vertices the pairs can be joined by edge-disjoint paths. The so far known families of path-pairable graphs have diameter of length at most 3. In this paper we present an…

In this work we prove the following: let $K$ be a convex body in the Euclidean space $\mathbb{R}^n$, $n\geq 3$, contained in the interior of the unit ball of $\mathbb{R}^n$, and let $p\in \mathbb{R}^n$ be a point such that, from each point…

Recent research on computing the diameter of geometric intersection graphs has made significant strides, primarily focusing on the 2D case where truly subquadratic-time algorithms were given for simple objects such as unit-disks and…

We consider a problem posed by Erd\H{o}s, Herzog and Piranian on the maximum product of distances of a point set of order $n$ with a given diameter. We prove that it is sufficient to consider convex polygons and obtain results on the…

In this paper we provide a method of finding possible numbers of shortest paths between two points in a space of compact sets in Euclidean space with Hausdorff distance. We also prove that there cannot be some of the numbers of shortest…