Related papers: Hyperplane Equipartitions Plus Constraints
The largest possible average diameter of a bounded cell of a simple hyperplane arrangement is conjectured to be not greater than the dimension. We prove that this conjecture holds in dimension 2, and is asymptotically tight in fixed…
In this paper, extending the recent work of authors with Calles Loperena and Dimitrijevi\'c Blagojevi\'c, we give a general and complete treatment of a problem of partition of mass assignments with prescribed arrangements of hyperplanes on…
Combinatorial optimization is a fertile testing ground for statistical physics methods developed in the context of disordered systems, allowing one to confront theoretical mean field predictions with actual properties of finite dimensional…
We consider the problem of jointly modeling and clustering populations of tensors by introducing a high-dimensional tensor mixture model with heterogeneous covariances. To effectively tackle the high dimensionality of tensor objects, we…
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…
We show that the minimal number of skewed hyperplanes that cover the hypercube $\{0,1\}^{n}$ is at least $\frac{n}{2}+1$, and there are infinitely many $n$'s when the hypercube can be covered with $n-\log_{2}(n)+1$ skewed hyperplanes. The…
Constrained clustering problems generalize classical clustering formulations, e.g., $k$-median, $k$-means, by imposing additional constraints on the feasibility of clustering. There has been significant recent progress in obtaining…
Uniformly distributed point sets on the unit sphere with and without symmetry constraints have been found useful in many scientific and engineering applications. Here, a novel variant of the Thomson problem is proposed and formulated as an…
We give an improved upper bound for the Gr\"unbaum--Hadwiger--Ramos problem: Let $d,n,k \in \mathbb{N}$ such that $d \geq 2^n(1+2^{k-1})$. Given $2^{n+1}$ masses on $\mathbb{R}^d$, there exist $k$ hyperplanes in $\mathbb{R}^d$ that…
A $\sqrt{n}$ estimate in the hyperplane problem with arbitrary measures has recently been proved in \cite{K3}. In this note we present analogs of this result for sections of lower dimensions and in the complex case. We deduce these…
Over the last two decades, pseudospectral methods based on Lagrange interpolants have flourished in solving trajectory optimization problems and their flight implementations. In a seemingly unjustified departure from these highly successful…
Many results in mass partitions are proved by lifting $\mathbb{R}^d$ to a higher-dimensional space and dividing the higher-dimensional space into pieces. We extend such methods to use lifting arguments to polyhedral surfaces. Among other…
Let $k$ be a field, $m$ a positive integer, $\mathbb{V}$ an affine subvariety of $\mathbb{A}^{m+3}$ defined by a linear relation of the form $x_{1}^{r_{1}}\cdots x_{m}^{r_{m}}y=F(x_{1}, \ldots , x_{m},z,t)$, $A$ the coordinate ring of…
We are interested in the study of parabolic equations on a multi-dimensional junction, i.e. the union of a finite number of copies of a half-hyperplane of dimension d + 1 whose boundaries are identified. The common boundary is referred to…
In this paper, we generalize classic mass partition results dealing with partitions using spheres, parallel hyperplanes, or axis-parallel wedges to the setting of mass assignments. In a mass assignment problem, we assign mass distributions…
We prove a version of the Erd\H{o}s--Beck Theorem from discrete geometry for fractal sets in all dimensions. More precisely, let $X\subset \mathbb{R}^n$ Borel and $k \in [0, n-1]$ be an integer. Let $\dim (X \setminus H) = \dim X$ for every…
A significant group of problems coming from the realm of Combinatorial Geometry can only be approached through the use of Algebraic Topology. From the first such application to Kneser's problem in 1978 by Lov% \'{a}sz \cite{Lovasz} through…
Contraction theory for dynamical systems on Euclidean spaces is well-established. For contractive (resp. semi-contractive) systems, the distance (resp. semi-distance) between any two trajectories decreases exponentially fast. For partially…
We consider a geometric combinatorial problem naturally associated to the geometric topology of certain spherical space forms. Given a collection of $m$ mass distributions on $\mathbb{R}^n$, the existence of $k$ affinely independent regular…
A problem that is frequently encountered in a variety of mathematical contexts, is to find the common invariant subspaces of a single, or set of matrices. A new method is proposed that gives a definitive answer to this problem. The key idea…