Related papers: On the multiple Borsuk numbers of sets

We give a short survey of problems and results on (1) diameter graphs and hypergraphs, and (2) geometric Ramsey theory. We also make some modest contributions to both areas. Extending a well known theorem of Kahn and Kalai which disproved…

We give the sharp lower bound of the volume product of three dimensional convex bodies which are invariant under a discrete subgroup of $O(3)$ in several cases. We also characterize the convex bodies with the minimal volume product in each…

The colourful simplicial depth problem in dimension d is to find a configuration of (d+1) sets of (d+1) points such that the origin is contained in the convex hull of each set (colour) but contained in a minimal number of colourful…

The problem of calculating exact lower bounds for the number of $k$-faces of $d$-polytopes with $n$ vertices, for each value of $k$, and characterising the minimisers, has recently been solved for $n\le2d$. We establish the corresponding…

Given natural numbers $k \leq s \leq n$, we ask: what is the minimal VC-dimension of a family $\mathcal{F}$ of $s$-subsets of $[n]$ that covers all $k$-subsets of $[n]$? We first show that for sufficiently large $n$ this number is always…

We consider a variation of the classical Erd\H{o}s-Szekeres problems on the existence and number of convex $k$-gons and $k$-holes (empty $k$-gons) in a set of $n$ points in the plane. Allowing the $k$-gons to be non-convex, we show bounds…

Let $s \in [0,1]$. We show that a Borel set $N \subset \mathbb{R}^{2}$ whose every point is linearly accessible by an $s$-dimensional family of lines has Hausdorff dimension at most $2 - s$.

The Euclidean minimum $M(K)$ of a number field $K$ is an important numerical invariant that indicates whether $K$ is norm-Euclidean. When $K$ is a non-CM field of unit rank 2 or higher, Cerri showed $M(K)$, as the supremum in the Euclidean…

Consider a set P of N random points on the unit sphere of dimension $d-1$, and the symmetrized set S = P union (-P). The halving polyhedron of S is defined as the convex hull of the set of centroids of N distinct points in S. We prove that…

We study the problem of representing all distances between $n$ points in $\mathbb R^d$, with arbitrarily small distortion, using as few bits as possible. We give asymptotically tight bounds for this problem, for Euclidean metrics, for…

We say that $E$ is a microset of the compact set $K\subset \mathbb{R}^d$ if there exist sequences $\lambda_n\geq 1$ and $u_n\in \mathbb{R}^d$ such that $(\lambda_n K + u_n ) \cap [0,1]^d$ converges to $E$ in the Hausdorff metric, and…

It is known that any periodic map of order $n$ on a closed oriented surface of genus $g$ can be equivariantly embedded into $S^m$ for some $m$. In the orientable and smooth category, we determine the smallest possible $m$ when $n\geq 3g$.…

The {\em minimal unfolded region} (or the {\em heart}) of a bounded subset $\Om$ in the Euclidean space is a subset of the convex hull of $\Om$ the definition of which is based on reflections in hyperplanes. It was introduced to restrict…

Thickenings of a metric space capture local geometric properties of the space. Here we exhibit applications of lower bounding the topology of thickenings of the circle and more generally the sphere. We explain interconnections with the…

We study the Borsuk-Ulam theorem for triple (M;\tau; \R^n), where M is a compact, connected, 3-manifold equipped with a fixed-point-free involution \tau. The largest value of n for which the Borsuk-Ulam theorem holds is called the Z_2-index…

This paper addresses the question: how should N n-dimensional subspaces of m-dimensional Euclidean space be arranged so that they are as far apart as possible? The results of extensive computations for modest values of N, n, m are…

Every graph G can be embedded in a Euclidean space as a two-distance set. The Euclidean representation number of G is the smallest dimension in which G is representable by such an embedding. We consider spherical and J-spherical…

We study the problem of high-dimensional multiple packing in Euclidean space. Multiple packing is a natural generalization of sphere packing and is defined as follows. Let $ N>0 $ and $ L\in\mathbb{Z}_{\ge2} $. A multiple packing is a set…

This thesis consists of five papers about reduced spherical convex bodies and in particular spherical bodies of constant width on the $d$-dimensional sphere $S^d$. In paper I we present some facts describing the shape of reduced bodies of…

We determine the set $S(d)$ of possible prime orders of $K$-rational points on elliptic curves over number fields $K$ of degree $d$, for $d = 4$, $5$, $6$, and $7$.