Related papers: Lacunary polynomials in $L^1$: geometry of the uni…

This work is a thorough and detailed study on the geometry of the unit sphere of certain Banach spaces of homogeneous polynomials in ${\mathbb{R}}^2$. Specifically, we provide a complete description of the unit spheres, identify the extreme…

It is an old question how massive polynomial hulls of Cantor sets in $\mathbb{C}^n$ can be. In contrast to expectation e.g. Rudin, Vitushkin and Henkin showed on examples that it can be rather massive. Motivated by problems of holomorphic…

We discuss the geometry of the unit ball -- specifically, the structure of its extreme points (if any) -- in subspaces of $L^1$ and $L^\infty$ on the circle that are formed by functions with prescribed spectral gaps. A similar issue is…

In this paper we discuss a couple of observations related to polynomial convexity. More precisely, (i) We observe that the union of finitely many disjoint closed balls with centres in $\cup_{\theta\in[0,\pi/2]}e^{i\theta}V$ is polynomially…

The decomposition of the polynomials on the quaternionic unit sphere in $\Hd$ into irreducible modules under the action of the quaternionic unitary (symplectic) group and quaternionic scalar multiplication has been studied by several…

Given any $\varepsilon>0$, we construct an orthonormal system of $n_k$ uniformly bounded polynomials of degree at most $k$ on the unit sphere in $\mathbb R^{m+1}$ where $n_k$ is bigger than $1-\varepsilon$ times the dimension of the space…

Any homogeneous polynomial $P(x, y, z)$ of degree $d$, being restricted to a unit sphere $S^2$, admits essentially a unique representation of the form $\lambda_0 + \sum_{k = 1}^d \lambda_k [\prod_{j = 1}^k L_{kj}]$, where $L_{kj}$'s are…

The notion of the higher rank numerical range $\Lambda_{k}(L(\lambda))$ for matrix polynomials $L(\lambda)=A_{m}\lambda^{m}+...+A_{1}\lambda+A_{0}$ is introduced here and some fundamental geometrical properties are investigated. Further,…

We obtain local and global results on polynomially convex hulls of Lagrangian and totally real submanifolds of $C^n$ with self-intersections and open Whitney umbrella points.

We discuss boundedness and compactness properties of the embedding $M_\Lambda^1\subset L^1(\mu)$, where $M_\Lambda^1$ is the closure of the monomials $x^{\lambda_n}$ in $L1([0,1])$ and $\mu$ is a finite positive Borel measure on the…

We study surfaces constructed from groups of units in quaternion orders $\Lambda$ over the integers in real quadratic fields k. A short presentation of some general theory of such surfaces is given, in particular, we construct certain…

Consider a random set of points on the unit sphere in $\mathbb{R}^d$, which can be either uniformly sampled or a Poisson point process. Its convex hull is a random inscribed polytope, whose boundary approximates the sphere. We focus on the…

Following the work of the first and last authors [2], we further analyze the structure of a zero set of a left ideal in the ring of central polynomials over the quaternion algebra H. We describe the "algebraic hull" of a point in H^n and…

By studying laminations of the unit disk, we can gain insight into the structure of Julia sets of polynomials and their dynamics in the complex plane. The polynomials of a given degree, $d$, have a parameter space. The hyperbolic components…

We study balls of homogeneous cubics on $\mathbb R^n$, $n = 2,3$, which are bounded by unity on the unit sphere. For $n = 2$ we completely describe the facial structure of this norm ball, while for $n = 3$ we classify all extremal points…

Given a subset $\Lambda$ of $\mathbb Z_+:=\{0,1,2,\dots\}$, let $H^\infty(\Lambda)$ denote the space of bounded analytic functions $f$ on the unit disk whose coefficients $\widehat f(k)$ vanish for $k\notin\Lambda$. Assuming that either…

We develop a constructive process which determines all extreme points of the unit ball of the space of $m$--linear forms, $m\geq1.$ Our method provides a full characterization of the geometry of that space through finitely many elementary…

In this work we prove the following: let $K$ be a convex body in the Euclidean space $\mathbb{R}^n$, $n\geq 3$, contained in the interior of the unit ball of $\mathbb{R}^n$, and let $p\in \mathbb{R}^n$ be a point such that, from each point…

Any homogeneous polynomial $P(x, y, z)$ of degree $d$, being restricted to a unit sphere $S^2$, admits essentially a unique representation of the form $\lambda + \sum_{k = 1}^d [\prod_{j = 1}^k L_{kj}]$, where $L_{kj}$'s are linear forms in…

We prove that every indefinite quadratic form with non-negative integer coefficients is the volume polynomial of a pair of lattice polygons. This solves the discrete version of the Heine-Shephard problem for two bodies in the plane. As an…