Related papers: Concrete constructions of non-pavable projections

It is known that the Kadison-Singer Problem (KS) and the Paving Conjecture (PC) are equivalent to the Bourgain-Tzafriri Conjecture (BT). Also, it is known that (PC) fails for $2$-paving projections with constant diagonal $1/2$. But the…

We recall two basic conjectures on the developables of convex projective curves, prove one of them and disprove the other in the firdt nontrivial case of curves in RP^3. Namely, we show that i) the tangent developable surface of any convex…

Let n be any integer greater than two. We prove that there exists a projection P having the following properties. (1) P is not the projection of any unknotted knot. (2) The singular point set of P consists of double points. (3) P is the…

We give a concrete example of a co-existential map between continua that is not confluent.

We study combinatorial configurations with the associated point and line graphs being strongly regular. Examples not belonging to known classes such as partial geometries and their generalizations or elliptic semiplanes are constructed.…

It is proved that the projection constants of two- and three-dimensional spaces are bounded by $4/3$ and $(1+\sqrt 5)/2$, respectively. These bounds are attained precisely by the spaces whose unit balls are the regular hexagon and…

It is well known that the Eisenbud-Goto regularity conjecture is true for arithmetically Cohen-Macaulay varieties, projective curves, smooth surfaces, smooth threefolds in $\mathbb{P}^5$, and toric varieties of codimension two. After J.…

We give some concrete constructions of real equiangular line sets. The emphasis here is on {\em building blocks} for certain angles which are then used to build up larger equiangular line sets. We concentrate on angles greater than or equal…

We consider the method of alternating projections for finding a point in the intersection of two closed sets, possibly nonconvex. Assuming only the standard transversality condition (or a weaker version thereof), we prove local linear…

Brandolini et al. conjectured that all concrete lattice polytopes can multitile the space. We disprove this conjecture in a strong form, by constructing an infinite family of counterexamples in $\mathbb{R}^3$.

We prove some constructive results that on first and maybe even on second glance seem impossible.

We prove some new equivalences of the paving conjecture and obtain some estimates on the paving constants. In addition we give a new family of counterexamples to one of the Akemann-Anderson conjectures.

Menger conjectured that subsets of R with the Menger property must be ${\sigma}$-compact. While this is false when there is no restriction on the subsets of R, for projective subsets it is known to follow from the Axiom of Projective…

As far as we know no notion of concreteness for fibrations exists. We introduce such a notion and discuss some basic results about it.

A projective rectangle is like a projective plane that may have different lengths in two directions. We develop properties of the graph of lines, in which adjacency means having a common point, especially its strong regularity and clique…

We give the first example of a mosaic of three combinatorial designs with distinct parameters $2$-$(13,3,1)$, $2$-$(13,4,2)$, and $2$-$(13,6,5)$. Furthermore, we give examples of mosaics of $2$-$(9,3,2)$ designs that are not resolvable,…

We consider Marstrand type projection theorems for closest-point projections in the normed space $\mathbb{R}^2$. We prove that if a norm on $\mathbb{R}^2$ is regular enough, then the analogues of the well-known statements from the Euclidean…

Two conjectures recently proposed by one of the authors are disproved

We prove a conjecture of Roe by constructing unified warped cones that violate the coarse Baum-Connes conjecture. Interestingly, the reason for this is probably not what Roe expected, as the obstruction arises in odd rather than even…

We characterize 1-complemented subspaces of finite codimension in strictly monotone one-$p$-convex, $2<p<\infty,$ sequence spaces. Next we describe, up to isometric isomorphism, all possible types of 1-unconditional structures in sequence…