Related papers: Plank theorems via successive inradii

In this paper, we consider the periodic tiling problem which was proved undecidable in the Euclidean plane by Yu. Gurevich and I. Koriakov in 1972. Here, we prove that the same problem for the hyperbolic plane is also undecidable.

The paper continues our previous work [7] on the radius of subregularity that was initiated by Asen Dontchev. We extend the results of [7] to general Banach/Asplund spaces and to other classes of perturbations, and sharpen the coderivative…

The major advances in physics have been through counterintuitive breakthroughs-- ideas that seemed to go against prevailing convictions. In the twentieth century the Special and General Theory of Relativity and Quantum Mechanics have…

The study of the topology of polynomial maps originates from classical questions in affine geometry, such as the Jacobian Conjecture, as well as from works of Whitney, Thom, and Mather in the 1950-70s on diffeomorphism types of smooth maps.…

The purpose of this paper is two-fold. We first prove a series of results, concerned with the notion of Zariski multiplicity, mainly for non-singular algebraic curves. These results are required in the paper "A Theory of Branches for…

In 1933, Borsuk proposed the following problem: Can every bounded set in $\mathbb{E}^n$ be divided into $n+1$ subsets of smaller diameters? This problem has been studied by many authors, and a lot of partial results have been discovered. In…

This is a glossary of notions and methods related with the topological theory of collections of affine planes, including braid groups, configuration spaces, order complexes, stratified Morse theory, simplicial resolutions, complexes of…

The pentagram map is a discrete integrable system on the moduli space of planar polygons. The corresponding first integrals are so-called monodromy invariants $E_1, O_1, E_2, O_2,\dots$ By analyzing the combinatorics of these invariants,…

This paper is devoted to demonstrate Kakeya's geometric proof of his theorem (1912), independently established earlier by Enestr\"om (1893). By calculating centers and radii of the interlacing circles of Kakeya's method, we prove Kakeya's…

We generalize the notion of Monk's schema in such a way to integrate finite dimensions. This allows us to lift a plathora of deep results proved for finite dimensions to the infinite dimensional case, like the solution to problem 2.12 in…

Given a convex set and an interior point close to the boundary, we prove the existence of a supporting hyperplane whose distance to the point is controlled, in a dimensionally quantified way, by the thickness of the convex set in the…

In this paper we prove the following theorem. Let $f$ be a dominant endomorphism of a smooth projective surface over an algebraically closed field of characteristic $0$. If there is no nonconstant invariant rational function under $f$, then…

The problem of classifying linear systems of conics in projective planes dates back at least to Jordan, who classified pencils (one-dimensional systems) of conics over $\mathbb{C}$ and $\mathbb{R}$ in 1906--1907. The analogous problem for…

Borsuk asked in 1933 if every set of diameter 1 in $R^d$ can be covered by $d+1$ sets of smaller diameter. In 1993, a negative solution, based on a theorem by Frankl and Wilson, was given by Kahn and Kalai. In this paper I will present…

Let $M$ be a $2$-space form. Let $P$ be a convex polygon in $M$. For these polygons, we define (and justify) a curvature $\kappa_i$ at each vertex $A_i$ of the polygon and and prove the following Blaschke's type theorem: If $P$ is a convex…

The Andreev-Thurston Circle Packing Theorem is generalized to packings of convex bodies in planar simply connected domains. This turns out to be a useful tool for constructing conformal and quasiconformal mappings with interesting geometric…

In this paper, we deal with the successive inner and outer radii with respect to Orlicz Minkowski sum. The upper and lower bounds for the radii of the Orlicz Minkowski sum of two convex bodies are established.

Using the formalism of Newton hyperplane arrangements, we resolve the open questions regarding angle rank left over from [DKRV20]. As a consequence we end up generalizing theorems of Lenstra--Zarhin and Tankeev proving several new cases of…

In a recent paper published in Nature, Y.I. Sobolev et al. introduced the concept of trajectoids: convex, rigid objects, which roll without slip or spin on a flat plane along a prescribed periodic, unbounded planar path. A geometric…

In 1925, Jarnik defined a sequence of convex polygons for use in constructing curves containing many lattice points relative to their curvatures. Properly scaled, these polygons converge to a certain limiting curve. In this paper we…