Related papers: Convex meets complex
Transport in crowded, complex environments occurs across many spatial scales. Geometric restrictions can hinder the motion of individuals and, combined with crowding between individuals, can have drastic effects on global transport…
Almost contact manifolds with B-metric are considered. Of special interest are the so-called vertical classes of the almost contact B-metric manifolds. Curvature properties of these manifolds are studied. An example of 5-dimensional…
A list of open problems on holomorphic symplectic, contact and Poisson manifolds.
We introduce a notion of $k$-convexity and explore polygons in the plane that have this property. Polygons which are \mbox{$k$-convex} can be triangulated with fast yet simple algorithms. However, recognizing them in general is a 3SUM-hard…
This paper deals with a generalized length-preserving flow for convex curves in the plane. It is shown that the flow exists globally and deforms convex curves into circles as time tends to infinity.
A correspondence between different $Pin$-type structures on a compact surface and quadratic (linear) forms on its homology is constructed. Addition of structures is defined and expressed in terms of these quadratic forms.
The correspondence between 2-parameter families of oriented lines in ${\Bbb{R}}^3$ and surfaces in $T{\Bbb{P}}^1$ is studied, and the geometric properties of the lines are related to the complex geometry of the surface. Congruences…
The usual notion of set-convexity, valid in the classical Euclidean context, metamorphoses into several distinct convexity types in the more general Riemannian setting. By studying this phenomenon in reverse, we characterize complete…
We give a detailed overview over known results for (no-)collision of a body with the boundary of its container.
One considers geometry with the intransitive equaivalence relation. Such a geometry is a physical geometry, i.e. it is described completely by the world function, which is a half of the squared distance function. The physical geometry…
We introduce notions of concavity for functions on balanced polyhedral spaces, and we show that concave functions on such spaces satisfy several strong continuity properties.
Consider the map $S$ which sends a planar polygon $P$ to a new polygon $S(P)$ whose vertices are the intersection points of second nearest sides of $P$. This map is the inverse of the famous pentagram map. In this paper we investigate the…
We present structures comprised of identical convex polyhedra which are interlocked geometrically. These sets cannot be disassembled by removing individual polyhedra by translations and/or rotations. The shapes that permit interlocking…
In this short survey article, we showcase a number of non-trivial geometric problems that have recently been resolved by marrying methods from functional calculus and real-variable harmonic analysis. We give a brief description of these…
3D shape creation and modeling remains a challenging task especially for novice users. Many methods in the field of computer graphics have been proposed to automate the often repetitive and precise operations needed during the modeling of…
The purpose of this note is to present several aspects of concentration phenomena in high dimensional geometry. At the heart of the study is a geometric analysis point of view coming from the theory of high dimensional convex bodies. The…
An attempt is made of giving a self-contained introduction to holomorphic ideas in general relativity, following work over the last thirty years by several authors. The main topics are complex manifolds, spinor and twistor methods, heaven…
It is well known that correlations predicted by quantum mechanics cannot be explained by any classical (local-realistic) theory. The relative strength of quantum and classical correlations is usually studied in the context of Bell…
We study some topological properties of attractors.
Solving a long-standing open question in convex geometry, we will show that typical convex surfaces contain points of infinite curvature in all tangent directions. To prove this, we use an easy curvature definition imitating the idea of…