Related papers: A new estimate of the transfinite diameter of Bern…

Given a compact set $K$ one may define a transfinite diameter for $K$ via a limiting process involving maximising a Vandermonde determinant over $K$ with respect to a monomial basis. Different transfinite diameters may be obtained by using…

We prove that if K is a remainder of the Hilbert space (i.e., K is the complement of the Hilbert space in its metrizable compactification) then every non-one-point closed image of K either contains a compact set with no transfinite…

We give a general formula for the $C-$transfinite diameter $\delta_C(K)$ of a compact set $K\subset \mathbb{C}^2$ which is a product of univariate compacta where $C\subset (\mathbb{R}^+)^2$ is a convex body. Along the way we prove a Rumely…

Let $0 \leq s \leq 1$. A set $K \subset \mathbb{R}^{2}$ is a Furstenberg $s$-set, if for every unit vector $e \in S^{1}$, some line $L_{e}$ parallel to $e$ satisfies $$\dim_{\mathrm{H}} [K \cap L_{e}] \geq s.$$ The Furstenberg set problem,…

For a Minkowski centered convex compact set $K$ we define $\alpha(K)$ to be the smallest possible factor to cover $K \cap (-K)$ by a rescalation of $\mathrm{conv} (K\cup (-K))$ and give a complete description of the possible values of…

We establish a higher-dimensional irrationality criterion for periods which are presented as Mellin integrals depending on many parameters. The criterion is stated as an upper bound on the multi-variate transfinite diameter of the image of…

We define $F$-polynomials as linear combinations of dilations by some frequencies of an entire function $F$. In this paper we use Pade interpolation of holomorphic functions in the unit disk by $F$-polynomials to obtain explicitly…

Given a porous compact $K \subset \mathbb{R}^d$ and a continuity modulus $\omega$, we prove a quantitative Jackson-Bernstein type theorem on harmonic approximation. That is, a function $f$ belongs to the class $\mathrm{Lip}_{\omega}(K)$ if…

Let $K\subset\mathbb{C}$ be a compact set in the plane whose logarithmic capacity $c(K)$ is strictly positive. Let $\mathscr{P}_n(K)$ be the space of monic polynomials of degree $n,$ \emph{all} of whose zeros lie in $K.$ For $p\in…

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…

We use methods from computational algebraic geometry to study Chebyshev constants and the transfinite diameter of a pure $m$-dimensional affine algebraic variety in $\mathbb{C}^n$ ($m\leq n$). The main result is a generalization of…

In this manuscript we introduce and study an extended version of the minimal dispersion of point sets, which has recently attracted considerable attention. Given a set $\mathscr P_n=\{x_1,\dots,x_n\}\subset [0,1]^d$ and…

We show that any compact subset of $\R^d$ which is the closure of a bounded star-shaped Lipschitz domain $\Omega$, such that $\complement \Omega$ has positive reach in the sense of Federer, admits an \emph{optimal AM} (admissible mesh),…

We study the minimal dimension of maximal commutative subalgebras of the matrix algebra $M_n(k)$ over an algebraically closed field. While examples with dimension strictly smaller than n are known for $n \geq 14$, no such examples are known…

We derive a new upper bound on the diameter of a polyhedron P = {x \in R^n : Ax <= b}, where A \in Z^{m\timesn}. The bound is polynomial in n and the largest absolute value of a sub-determinant of A, denoted by \Delta. More precisely, we…

Given a subset K of the unit Euclidean sphere, we estimate the minimal number m = m(K) of hyperplanes that generate a uniform tessellation of K, in the sense that the fraction of the hyperplanes separating any pair x, y in K is nearly…

Let $K$ be an algebraically closed field and let $M_n(K)$ denote the algebra of $n\times n$ matrices over $K$. A classical problem asks for the minimal possible dimension of a maximal commutative subalgebra $A \subseteq M_n(K)$. We…

We completely describe the region of possible values of the diameter-width ratio for planar pseudo-complete sets in dependence of the Minkowski asymmetry. In order to do this, we focus on the containment inequalities of $K \cap (-K)$ and…

Let $f$ be an analytic polynomial of degree at most $K-1$. A classical inequality of Bernstein compares the supremum norm of $f$ over the unit circle to its supremum norm over the sampling set of the $K$-th roots of unity. Many extensions…

For the sequence of multi-indexes $\{\alpha_i\}_{i=1}^{m}$ and $\beta$ we study the inequality \[ \|D^{\beta} f\|_{L_1(\mathbb{T}^d)}\leq K_N \sum_{j= 1}^{m} \|D^{\alpha_j}f\|_{L_1(\mathbb{T}^d)}, \] where $f$ is a trigonometric polynomial…