Related papers: Large-volume open sets in normed spaces without in…
Previous studies on the geometrical properties of the state space of a finite-level quantum system have determined its volume and surface area. Building on this foundation, we derive explicit formulas for two additional intrinsic volume…
Let $S$ be a set of $n$ points in $\mathbb{R}^d$. A Steiner convex partition is a tiling of ${\rm conv}(S)$ with empty convex bodies. For every integer $d$, we show that $S$ admits a Steiner convex partition with at most $\lceil…
We consider two disjoint sets of points. If at least one of the sets can be embedded into an Euclidean space, then we provide sufficient conditions for the two sets to be jointly embedded in one Euclidean space. In this joint Euclidean…
The aim of this paper is to present a survey of some recent results obtained in the study of spaces with asymmetric norm. The presentation follows the ideas from the theory of normed spaces (topology, continuous linear operators, continuous…
In this paper we develop some combinatorial models for continuous spaces. In this spirit we study the approximations of continuous spaces by graphs, molecular spaces and coordinate matrices. We define the dimension on a discrete space by…
The problem of estimating, from a random sample of points, the dimension of a compact subset $S$ of the Euclidean space is considered. The emphasis is put on consistency results in the statistical sense. That is, statements of convergence…
Periodic point sets model all solid crystalline materials (crystals) whose atoms can be considered zero-sized points with or without atomic types. This paper addresses the fundamental problem of checking whether claimed crystals are novel,…
Bounding volumes are an established concept in computer graphics and vision tasks but have seen little change since their early inception. In this work, we study the use of neural networks as bounding volumes. Our key observation is that…
In this paper, we mainly introduce the notion of an open uniform (G) at non-isolated points, and show that a space $X$ has an open uniform (G) at non-isolated points if and only if $X$ is the open boundary-compact image of metric spaces.…
Let X be a normed linear space. We examine if every open, convex and unbounded subset of X is equal to the union of a family of open straight half lines. The answer is affirmative if and only if X is finite dimensional.
We extend an implicit regularization scheme to be applicable in the $n$-dimensional space-time. Within this scheme divergences involving parity violating objects can be consistently treated without recoursing to dimensional continuation.…
Volume polynomials form a distinguished class of log-concave polynomials with remarkable analytic and combinatorial properties. I will survey realization problems related to them, review fundamental inequalities they satisfy, and discuss…
A homogeneous set of $n$ points in the $d$-dimensional Euclidean space determines at least $\Omega(n^{2d/(d^2+1)} / \log^{c(d)} n)$ distinct distances for a constant $c(d)>0$. In three-space, we slightly improve our general bound and show…
We use an entropy based method to study two graph maximization problems. We upper bound the number of matchings of fixed size $\ell$ in a $d$-regular graph on $N$ vertices. For $\frac{2\ell}{N}$ bounded away from 0 and 1, the logarithm of…
The pattern avoidance problem seeks to construct a set $X\subset \mathbb{R}^d$ with large dimension that avoids a prescribed pattern. Examples of such patterns include three-term arithmetic progressions (solutions to $x_1 - 2x_2 + x_3 =…
In this paper, we consider a certain variation of the "isoperimetric problem" adopted for subsets of nonnegative integers. More specifically, we explore the sequence P(n) as described in OEIS A186053. We provide the first exact formulas for…
Motivated by applications in robotics and computer vision, we study problems related to spatial reasoning of a 3D environment using sublevel sets of polynomials. These include: tightly containing a cloud of points (e.g., representing an…
In the maximum independent set of convex polygons problem, we are given a set of $n$ convex polygons in the plane with the objective of selecting a maximum cardinality subset of non-overlapping polygons. Here we study a special case of the…
In this paper, we study some topological characteristics of the n-normed spaces. We observe convergence sequences, closed sets, and bounded sets in the n-normed spaces using norms of quotient spaces that will be constructed. These norms…
We prove a limit theorem for the the maximal interpoint distance (also called the diameter) for a sample of n i.i.d. points in the unit ball of dimension 2 or more. The exact form of the limit distribution and the required normalisation are…