Related papers: Arithmetic on Moran sets

If $C_n(\mathbb{R}^d)$ denotes the configuration space of $n$ distinct points in $\mathbb{R}^d$, we construct a sequence of maps $(f_m),$ $m \geq 1$, where \[f_m: C_n(\mathbb{R}^d) \times \mathbb{R}^d \to \mathbb{R}^d\] is real analytic,…

We consider regular endomorphisms of the complex affine space with a degree gap $k$. They are endomorphisms $f$ of $\mathbb{A}_{\mathbb{C}}^{N}$ of the form…

In this article a class of closed convex sets in the Euclidean $n$-space which are the convex hull of their profiles is described. Thus a generalization of Krein-Milman theorem\cite{Lay:1982} to a class of closed non-compact convex sets is…

A compact set $E\subset {\Bbb R}^d$ is said to be arithmetically thick if there exists a positive integer $n$ so that the $n$-fold arithmetic sum of $E$ has non-empty interior. We prove the arithmetic thickness of $E$, if $E$ is uniformly…

For a complex parameter $c$ outside the unit disk and an integer $n\ge2$, we examine the $n$-ary collinear fractal $E(c,n)$, defined as the attractor of the iterated function system $\{\mbox{$f_k \colon \mathbb{C} \longrightarrow…

The shape of homogeneous, generic, smooth convex bodies as described by the Euclidean distance with nondegenerate critical points, measured from the center of mass represents a rather restricted class M_C of Morse-Smale functions on S^2.…

Let $\alpha_1, \ldots, \alpha_m$ be two or more positive reals with sum $1$, let $C\subseteq \mathbb{R}^k$ be an open convex set, and $f: C\to \mathbb{R}^k$ be a continuous injection with convex image. For each nonempty set $S\subseteq C$,…

Suppose that $f$ belongs to a suitably defined complete metric space $ {{\cal C}}^{{\alpha}}$ of H\"older $ {\alpha}$-functions defined on $[0,1]$. We are interested in whether one can find large (in the sense of Hausdorff, or lower/upper…

In this note we examine the volume of the convex hull of two congruent copies of a convex body in Euclidean $n$-space, under some subsets of the isometry group of the space. We prove inequalities for this volume if the two bodies are…

In this paper, we consider compact free boundary constant mean curvature surfaces immersed in a mean convex body of the Euclidean space or in the unit sphere. We prove that the Morse index is bounded from below by a linear function of the…

Let $S$ be a convex hypersurface (the boundary of a closed convex set $V$ with nonempty interior) in $\mathbb{R}^n$. We prove that $S$ contains no lines if and only if for every open set $U\supset S$ there exists a real-analytic convex…

Let (N,g) be a Riemannian manifold. For a compact, connected and oriented submanifold M of N. we define the space of volume preserving embeddings Emb_{\mu}(M,N) as the set of smooth embeddings f:M \rightarrow N such that f*\mu^{f}=\mu,…

Let $A(K)$ be the algebra of continuous functions on a compact set $K\subset\mathbb C$ which are analytic on the interior of $K$, and $R(K)$ the closure (with the uniform convergence on $K$) of the functions that are analytic on a…

Cartan-Thullen theorem is a basic one in the theory of analytic functions of several complex variables. It states that for any open set $U$ of ${\mathbb C}^k$, the following conditions are equivalent: (a) $U$ is a domain of existence, (b)…

Let $X$ be a Borel subset of the Cantor set \textbf{C} of additive or multiplicative class ${\alpha},$ and $f: X \to Y$ be a continuous function with compact preimages of points onto $Y \subset \textbf{C}.$ If the image $f(U)$ of every…

We study the following open problem, suggested by Barker and Larman. Let $K$ and $L$ be convex bodies in $\mathbb R^n$ ($n\ge 2$) that contain a Euclidean ball $B$ in their interiors. If $\mathrm{vol}_{n-1}(K\cap H) =…

Let $E_1,\dots,E_n \subset \mathbb{R}^d$ be compact sets of positive diameter with Feng--Wu thickness at least $c>0$. Feng and Wu proved that $E_1+\cdots+E_n$ has non-empty interior when $n>2^{11}c^{-3}+1$. We show that \[n>\frac{\sqrt…

Let $K$ be a compact convex set and $m$ be a positive integer. The covering functional of $K$ with respect to $m$ is the smallest $\lambda\in[0,1]$ such that $K$ can be covered by $m$ translates of $\lambda K$. Estimations of the covering…

It is shown that a simple closed curve in $\mathbb C^n$ that is a uniform limit of rectifiable simple closed curves each of which has nontrivial polynomial hull has itself nontrivial polynomial hull. In case the limit curve is rectifiable,…

Let $M$ be a convex body and let $K$ be a closed convex surface $K$ both contained in the Euclidean space $\mathbb{E}^3$. What can we say about $M$ if $K$ encloses $M$ and if from all the points in $K$ the body $M$ looks the same? In this…