Related papers: A new discrete theory of pseudoconvexity

In a series of previous papers, we initiated a systematic study of semihypergroups and had a thorough discussion on certain analytic and algebraic aspects associated to this class of objects. In particular, we introduced the notion of…

Let $p$ be a real zero polynomial in $n$ variables. Then $p$ defines a rigidly convex set $C(p)$. We construct a linear matrix inequality of size $n+1$ in the same $n$ variables that depends only on the cubic part of $p$ and defines a…

Semigraphoids are combinatorial structures that arise in statistical learning theory. They are equivalent to convex rank tests and to polyhedral fans that coarsen the reflection arrangement of the symmetric group. We resolve two problems on…

The theory of limits of discrete combinatorial objects has been thriving for the last decade or so. The syntactic, algebraic approach to the subject is popularly known as "flag algebras", while the semantic, geometric one is often…

We primarily investigate the properties of characteristic polynomials of semimatroids. In particular, we provide a combinatorial interpretation of their coefficients, generalizing the Whitney's Broken Circuit Theorem. We also prove that the…

Let T be the unit circle in the complex plane C. This paper proves the existence of analytic structure in a compact subset K of T X C^n, where K has so-called "lineally convex" or "hypoconvex" fibers over T. It also addresses a related…

The symmetries described by Pin groups are the result of combining a finite number of discrete reflections in (hyper)planes. The current work shows how an analysis using geometric algebra provides a picture complementary to that of the…

In the present paper, we show that many combinatorial and topological objects, such as maps, hypermaps, three-dimensional pavings, constellations and branched coverings of the two--sphere admit any given finite automorphism group. This…

We develop a theory for the existence of perfect matchings in hypergraphs under quite general conditions. Informally speaking, the obstructions to perfect matchings are geometric, and are of two distinct types: 'space barriers' from convex…

Let $C_1,...,C_{d+1}$ be $d+1$ point sets in $\mathbb{R}^d$, each containing the origin in its convex hull. A subset $C$ of $\bigcup_{i=1}^{d+1} C_i$ is called a colorful choice (or rainbow) for $C_1, \dots, C_{d+1}$, if it contains exactly…

We provide a new and elegant approach to relative quasiconvexity for relatively hyperbolic groups in the context of Bowditch's approach to relative hyperbolicity using cocompact actions on fine hyperbolic graphs. Our approach to…

This paper derives central limit and bootstrap theorems for probabilities that sums of centered high-dimensional random vectors hit hyperrectangles and sparsely convex sets. Specifically, we derive Gaussian and bootstrap approximations for…

Swartz proved that any matroid can be realized as the intersection lattice of an arrangement of codimension one homotopy spheres on a sphere. This was an unexpected extension from the oriented matroid case, but unfortunately the…

We prove a common generalization to several mass partition results using hyperplane arrangements to split $\mathbb{R}^d$ into two sets. Our main result implies the ham-sandwich theorem, the necklace splitting theorem for two thieves, a…

In Guo and Peng's article [Spherically convex sets and spherically convex functions, J. Convex Anal. 28 (2021), 103--122], one defines the notions of spherical convex sets and functions on "general curved surfaces" in $\mathbb{R}^{n}$…

The main theme of the paper is the dynamics of Hamiltonian diffeomorphisms of ${\mathbb C}{\mathbb P}^n$ with the minimal possible number of periodic points (equal to $n+1$ by Arnold's conjecture), called here Hamiltonian pseudo-rotations.…

We use techniques from algebraic and extremal combinatorics to derive upper bounds on the number of independent sets in several (hyper)graphs arising from finite geometry. In this way, we obtain asymptotically sharp upper bounds for partial…

We introduce several classes of set-valued maps with generalized convexity. We obtain minimax theorems for set-valued maps which satisfy the introduced properties and are not continuous, by using a fixed point theorem for weakly naturally…

The paper is the survey of the modern results and applications of the theory of homotopes. The notion of a well-tempered element in an associative algebra is introduced and it is proven that the category of representations of the homotope…

In this paper, two parallel notions of convexity of sets are introduced in the abelian semigroup setting. The connection of these notions to algebraic and to set-theoretic operations is investigated. A formula for the computation of the…