Related papers: Helly's Theorem: New Variations and Applications
We raise a question related to Helly's theorem with the added elements of geometric transformations.
The main results here are two Helly type theorems for the sum of (at most) unit vectors in a normed plane. Also, we give a new characterization of centrally symmetric convex sets in the plane.
Helly's theorem is a classical result concerning the intersection patterns of convex sets in $\mathbb{R}^d$. Two important generalizations are the colorful version and the fractional version. Recently, B\'{a}r\'{a}ny et al. combined the…
In this paper, we present a variety of problems in the interface between combinatorics and geometry around the theorems of Helly, Radon, Carath\'eodory, and Tverberg. Through these problems we describe the fascinating area of Helly-type…
In this paper we will survey some recent developments in the last decade or so on variation of Geometric Invariant Theory and its applications to Birational Geometry such as the weak Factorization Theorems of nonsingular projective…
We discuss recent progress on topological Helly-type theorems and their variants. We provide an overview of two different proof techniques, one based on the nerve lemma, while the other on non-embeddability.
We study versions of Helly's theorem that guarantee that the intersection of a family of convex sets in $R^d$ has a large diameter. This includes colourful, fractional and $(p,q)$ versions of Helly's theorem. In particular, the fractional…
We propose an interpretation of, and approach to, Helly's theorem that can be included quite early in the undergraduate curriculum. At the same time, the approach connects with contemporary models of data privacy and with sampling methods…
Families of translates and homothets of strictly convex curves are proven to possess Helly-type properties generalizing those of a circle. Weaker results are shown for arbitrary convex curves.
We survey the development of Clifford's geometric algebra and some of its engineering applications during the last 15 years. Several recently developed applications and their merits are discussed in some detail. We thus hope to clearly…
We survey a number of Weyl type laws that have recently been established in low-dimensional symplectic geometry. These have had a number of applications, which we also introduce. We sketch a number of proofs so that the reader can get a…
This is a brief review, in relatively non-technical terms, of recent advances in the theory of random field geometry. These advances have provided a collection of explicit new formulae describing mean values of a variety of geometric…
This paper provides an overview of modern digital geometry and topology through mathematical principles, algorithms, and measurements. It also covers recent developments in the applications of digital geometry and topology including image…
This is a survey article on real algebra and geometry, and in particular on its recent applications in optimization and convexity. We first introduce basic notions and results from the classical theory. We then explain how these relate to…
The paper suggests a short survey of integration algorithms which evolved since 1982. These theorems and algorithms form discrete versions of the calculus theorems.
We introduce and study a new class of $\eps$-convex bodies (extending the class of convex bodies) in metric and normed linear spaces. We analyze relations between characteristic properties of convex bodies, demonstrate how $\eps$-convex…
We give a complete geometrical description of the effective Hamiltonians common in nuclear shell model calculations. By recasting the theory in a manifestly geometric form, we reinterpret and clarify several points. Some of these results…
This is a survey of vanishing and positivity theorems for Hodge modules, and their recent applications to birational and complex geometry, expanding on my lecture at the 2015 AMS Summer Institute.
We report on some recent progress regarding combinatorial properties in convexity spaces with a bounded Radon number. In particular, we discuss the relationship between the Radon number, the colorful and fractional Helly properties, weak…
In this article we will represent some ideas and a lot of new theorems in Euclidean plane geometry.