相关论文: Tilings and coverings by balls in $\ell_1$
We construct a bounded and symmetric convex body in $\ell_2(\Gamma)$ (for certain cardinals $\Gamma$) whose translates yield a tiling of $\ell_2(\Gamma)$. This answers a question due to Fonf and Lindenstrauss. As a consequence, we obtain…
In 1989, G. Godefroy proved that a Banach space contains an isomorphic copy of $\ell_1$ if and only if it can be equivalently renormed to be octahedral. It is known that octahedral norms can be characterized by means of covering the unit…
We adapt a construction of Klee (1981) to find a packing of unit balls in $\ell_p$ ($1\leq p<\infty$) which is efficient in the sense that enlarging the radius of each ball to any $R>2^{1-1/p}$ covers the whole space. We show that the value…
We show some new results about tilings in Banach spaces. A tiling of a Banach space $X$ is a covering by closed sets with non-empty interior so that they have pairwise disjoint interiors. If moreover the tiles have inner radii uniformly…
The main result of this paper is to show that, if $\kappa$ is the smallest real-valued measurable cardinal not greater than $ 2^{\aleph_0}$, then there exists a complete metric space of cardinality not greater than $ 2^{\kappa}$ admitting a…
Let kappa be an uncountable cardinal and the edges of a complete graph with kappa vertices be colored with aleph_0 colors. For kappa >2^{aleph_0} the Erd\H{o}s-Rado theorem implies that there is an infinite monochromatic subgraph. However,…
In this note, we provide a starlike and normal tiling in any separable Banach space. That means, there are positive constants r and R (not depending on the separable Banach space) such that every tile of this tiling is starlike, contains a…
We introduce and investigate a class of non-separable tree-like Banach spaces. As a consequence, we prove that we can not achieve a satisfactory extension of Rosenthal's $\ell_1$-theorem to spaces of the type $\ell_1(\kappa)$, for $\kappa$…
We answer in the affirmative the following question raised by H. H. Corson in 1961: "Is it possible to cover every Banach space X by bounded convex sets with nonempty interior in such a way that no point of X belongs to infinitely many of…
We study star-finite coverings of infinite-dimensional normed spaces. A family of sets is called star-finite if each of its members intersects only finitely many other members of the family. It follows by our results that an LUR or a…
We prove that the existence of a complete metric space of cardinality at most $2^{\kappa}$ admitting Kuratowski partition is a consequence of $\kappa$ being the smallest real-valued measurable cardinal not greater than $ 2^{\aleph_0}$.
We provide an elementary proof of a result by V.P.~Fonf and C.~Zanco on point-finite coverings of separable Hilbert spaces. Indeed, by using a variation of the famous argument introduced by J.~Lindenstrauss and R.R.~Phelps \cite{LP} to…
Given a Banach space we consider the $\sigma$-ideal of all of its subsets which are covered by countably many hyperplanes and investigate its standard cardinal characteristics as the additivity, the covering number, the uniformity, the…
R. Nandakumar asked whether there is a tiling of the plane by pairwise incongruent triangles of equal area and equal perimeter. Recently a negative answer was given by Kupavskii, Pach and Tardos. Still one may ask for weaker versions of the…
A result of G. Godefroy asserts that a Banach space $X$ contains an isomorphic copy of $\ell_1$ if and only if there is an equivalent norm $|||\cdot|||$ such that, for every finite-dimensional subspace $Y$ of $X$ and every $\varepsilon>0$,…
We continue our study of Sierpinski-type colourings. In contrast to the prequel paper, we focus here on colourings for ideals stratified by their completeness degree. In particular, improving upon Ulam's theorem and its extension by Hajnal,…
For any delta > 1 we construct a periodic and locally finite packing of the plane with ellipses whose delta-enlargement covers the whole plane. This answers a question of Imre B\'ar\'any. On the other hand, we show that if C is a packing in…
In [BNRR], it was shown that tiling of general regions with two rectangles is NP-complete, except for a few trivial special cases. In a different direction, R\'emila showed that for simply connected regions by two rectangles, the…
Let $\kappa$ be a regular cardinal. Consider the Baire numbers of the spaces $(2^{\theta})_\kappa$ (functions from $\theta$ to 2 and the less than $\kappa$ topology) for various $\theta \geq \kappa$. Let l be the number of such different…
Say that a cardinal number $\kappa$ is \emph{small} relative to the space $X$ if $\kappa <\Delta(X)$, where $\Delta(X)$ is the least cardinality of a non-empty open set in $X$. We prove that no Baire metric space can be covered by a small…