Related papers: An Approach to the Hirsch Conjecture
In this paper we adopt a geometric point of view regarding a famous conjecture due to Littlewood in diophantine approximation of real numbers. Following the spirit of the geometric theory of continued fractions, we give a sufficient…
We begin the study of the notion of diameter of an ideal I of a polynomial ring S over a field, an invariant measuring the distance between the minimal primes of I. We provide large classes of Hirsch ideals, i.e. ideals with diameter not…
Some conjectures and open problems in convex geometry are presented, and their physical origin, meaning, and importance, for quantum theory and generic statistical theories, are briefly discussed.
We prove a conjecture of Goncharov, which says that any multiple polylogarithm can be expressed via polylogarithms of depth at most half of the weight. We give an explicit formula for this presentation, involving a summation over trees that…
A very fundamental geometric problem on finite systems of spheres was independently phrased by Kneser (1955) and Poulsen (1954). According to their well-known conjecture if a finite set of balls in Euclidean space is repositioned so that…
Polytopes are one of the most primitive concepts underlying geometry. Discovery and study of polytopes with complex structures provides a means of advancing scientific knowledge. Construction of polytopes with specific extremal structure is…
The volume of a Meissner polyhedron is computed in terms of the lengths of its dual edges. This allows to reformulate the Meissner conjecture regarding constant width bodies with minimal volume as a series of explicit finite dimensional…
We prove that every polytope described by algebraic coordinates is the face of a projectively unique polytope. This provides a universality property for projectively unique polytopes. Using a closely related result of Below, we construct a…
Let $(M, \dr M)$ be a 3-manifold with incompressible boundary that admits a convex co-compact hyperbolic metric. We consider the hyperbolic metrics on $M$ such that $\dr M$ looks locally like a hyperideal polyhedron, and we characterize the…
This article is devoted to examples of (orbifold) K\"ahler groups from the perspective of the so-called Shafarevich conjecture on holomorphic convexity. It aims at pointing out that every quasi-projective complex manifold with an…
We present a much simplified proof of Dehn's theorem on the infinitesimal rigidity of convex polytopes. Our approach is based on the ideas of Trushkina and Schramm.
We prove that the intersection of a Hirsch polytope and a cube may be a non-Hirsch polytope.
We establish the second part of Milnor's conjecture on the volume of simplexes in hyperbolic and spherical spaces. A characterization of the closure of the space of the angle Gram matrices of simplexes is also obtained.
Santos' construction of the first known counterexample to the Hirsch conjecture, for bounded polytopes, follows the strategy of first finding a counterexample to the nonrevisiting conjecture. Santos constructs a $5$-dimensional…
The notion of Ehrhart tensor polynomials, a natural generalization of the Ehrhart polynomial of a lattice polytope, was recently introduced by Ludwig and Silverstein. We initiate a study of their coefficients. In the vector and matrix…
First, we shall formulate and prove Theorem of Lie-Kolchin type for a cone and derive some algebro-geometric consequences. Next, inspired by a recent result of Dinh and Sibony we pose a conjecture of Tits type for a group of automorphisms…
In [7], Higashitani, Kummer, and Micha{\l}ek pose a conjecture about the symmetric edge polytopes of complete multipartite graphs and confirm it for a number of families in the bipartite case. We confirm that conjecture for a number of new…
In ["Illumination of convex bodies with many symmetries", Mathematika 63 (2017)], Tikhomirov verified the Hadwiger-Boltyanski Illumination Conjecture for the class of 1-symmetric convex bodies of sufficiently large dimension. We propose an…
In this paper we give a proof of the Manickam-Mikl\'os-Singhi (MMS) conjecture for some partial geometries. Specifically, we give a condition on partial geometries which implies that the MMS conjecture holds. Further, several specific…
Beginning from the resolution of the Dirichlet L function, using the inner product formula between two infinite-dimensional vectors in the complex space, the author proved the baffling problem--Hecke conjecture.