Related papers: On the multiple Borsuk numbers of sets

A subset of a metric space is a k-distance set if there are exactly k non-zero distances occuring between points. We conjecture that a k-distance set in a d-dimensional Banach space (or Minkowski space), contains at most (k+1)^d points,…

Borsuk's conjecture states that any bounded set in R^n can be partitioned into n+1 sets of smaller diameter. It is known to be false for all n bigger or equal to 323. Here we show that Borsuk's conjecture fails in dimensions 321 and 322.…

A generalization of the Borsuk-Ulam theorem to Stiefel manifolds is considered. This theorem is applied to derive bounds on $d$ that guarantee-for a given set of $m$ measures in $\mathbb{R}^d$-the existence of $k$ mutually orthogonal…

We investigate the behavior of small subsets of causal sets that approximate Minkowski space in three, four, and five dimensions, and show that their effective dimension decreases smoothly at small distances. The details of the short…

Let $\mathcal{O}_K$ be the ring of integers of an algebraic number field $K$ embedded into $\mathbb{C}$. Let $X$ be a subset of the Euclidean space $\mathbb{R}^d$, and $D(X)$ be the set of the squared distances of two distinct points in…

We study the smallest intersecting and enclosing ball problems in Euclidean spaces for input objects that are compact and convex. They link and unify many problems in computational geometry and machine learning. We show that both problems…

Let $L_{mkl}\subset \mathbb{R}^{m+k+l}$ be the set of vectors which have $m$ of entries $-1$, $k$ of entries $0$, and $l$ of entries $1$. In this paper, we investigate the largest subset of $L_{mkl}$ whose diameter is smaller than that of…

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…

The minimal spherical cap dispersion ${\rm disp}_{\mathcal{C}}(n,d)$ is the largest number $\varepsilon\in (0,1]$ such that, for every $n$ points on the $d$-dimensional Euclidean unit sphere $\mathbb{S}^d$, there exists a spherical cap with…

A simple n-gon is a polygon with n edges such that each vertex belongs to exactly two edges and every other point belongs to at most one edge. Brass, Moser and Pach asked the following question: For n > 3 odd, what is the maximum perimeter…

Let ||.|| be a norm in R^d whose unit ball is B. Assume that V\subset B is a finite set of cardinality n, with \sum_{v \in V} v=0. We show that for every integer k with 0 \le k \le n, there exists a subset U of V consisting of k elements…

The lattice diameter of a bounded set $S \subset \mathbb{R}^d$ measures the maximal number of lattice points in a segment whose endpoints are lattice points in $S$. Such a segment is called a lattice diameter segment of $S$. This simple…

If K' and K are convex bodies of the plane such that K' is a subset of K then the perimeter of K' is not greater than the perimeter of K. We obtain the following generalization of this fact. Let K be a convex compact body of the plane with…

The purpose of this article is to demonstrate the connection between the properties of the Gromov--Hausdorff distance and the Borsuk conjecture. The Borsuk number of a given bounded metric space $X$ is the infimum of cardinal numbers $n$…

The interrelations between (upper and lower) Minkowski contents and (upper and lower) surface area based contents (S-contents) as well as between their associated dimensions have recently been investigated for general sets in R^d (cf. [3]).…

Given n points in Euclidean space E^d, we propose an algebraic algorithm to compute the best fitting (d-1)-cylinder. This algorithm computes the unknown direction of the axis of the cylinder. The location of the axis and the radius of the…

We study the complexity of geometric problems on spaces of low fractal dimension. It was recently shown by [Sidiropoulos & Sridhar, SoCG 2017] that several problems admit improved solutions when the input is a pointset in Euclidean space…

We investigate how the Hausdorff dimensions of microsets are related to the dimensions of the original set. It is known that the maximal dimension of a microset is the Assouad dimension of the set. We prove that the lower dimension can…

Our aim is to find the minimal Hausdorff dimension of the union of scaled and/or rotated copies of the $k$-skeleton of a fixed polytope centered at the points of a given set. For many of these problems, we show that a typical arrangement in…

We derive a new estimate of the size of finite sets of points in metric spaces with few distances. The following applications are considered: (1) we improve the Ray-Chaudhuri--Wilson bound of the size of uniform intersecting families of…