Related papers: Rational Maps and Boundaries of Convex Hulls

A \emph{thrackle} is a graph drawn in the plane so that every pair of its edges meet exactly once, either at a common end vertex or in a proper crossing. Conway's thrackle conjecture states that the number of edges is at most the number of…

Let $C$ be a proper convex cone generated by a compact set which supports a measure $\mu$. A construction due to A.Barvinok, E.Veomett and J.B. Lasserre produces, using $\mu$, a sequence $(P_k)_{k\in \mathbb{N}}$ of nested spectrahedral…

Convex hulls are a fundamental geometric tool used in a number of algorithms. As a side-effect of exhaustive tests for an algorithm for which a convex hull computation was the first step, interesting experimental results were found and are…

Jensen's inequality is ubiquitous in measure and probability theory, statistics, machine learning, information theory and many other areas of mathematics and data science. It states that, for any convex function $f\colon K \to \mathbb{R}$…

Magnitude of a finite metric space and the related notion of magnitude functions on metric spaces is an active area of research in algebraic topology. Magnitude originally arose in the context of biology, where it represents the number of…

We consider ensembles of planar maps with two marked vertices at distance k from each other and look at the closed line separating these vertices and lying at distance d from the first one (d<k). This line divides the map into two…

Let $X_i = {X_i(t), t \in T}$ be i.i.d. copies of a centered Gaussian process $X = {X(t), t \in T}$ with values in $\mathbb{R}^d$ defined on a separable metric space $T.$ It is supposed that $X$ is bounded. We consider the asymptotic…

We prove (and improve) the Muir-Suffridge conjecture for holomorphic convex maps. Namely, let $F:\mathbb B^n\to \mathbb C^n$ be a univalent map from the unit ball whose image $D$ is convex. Let $\mathcal S\subset \partial \mathbb B^n$ be…

We construct a locally compact Hausdorff topology on the path space of a finitely aligned $k$-graph $\Lambda$. We identify the boundary-path space $\partial\Lambda$ as the spectrum of a commutative $C^*$-subalgebra $D_\Lambda$ of…

Let $U\subseteq\mathbb{R}^d$ be open and convex. We prove that every (not necessarily Lipschitz or strongly) convex function $f:U\to\mathbb{R}$ can be approximated by real analytic convex functions, uniformly on all of $U$. We also show…

Writing an uncomplicated, robust, and scalable three-dimensional convex hull algorithm is challenging and problematic. This includes, coplanar and collinear issues, numerical accuracy, performance, and complexity trade-offs. While there are…

Let $D\subset \mathbb{R}^{n}$ be a bounded convex domain and $F:D\rightarrow D$ a $1$-Lipschitz mapping with respect to the Hilbert metric $d$ on $D$ satisfying condition $d(sx+(1-s)y,sz+(1-s)w)\leq \max \{d(x,z),d(y,w) \}$. We show that if…

We prove that the combinatorial optimization problem of determining the hull number of a partial cube is NP-complete. This makes partial cubes the minimal graph class for which NP-completeness of this problem is known and improves some…

A characterization of the proximal normal cone is obtained and a separation theorem for convex subsets of Riemannian manifolds is established. Moreover, the convexity of the distance function $d_S$ for a convex subset $S$ in the cases where…

In this paper, we characterize the closures of convex hulls of unitary orbits of self-adjoint operators in unital, separable, simple C$^*$-algebras with non-trivial tracial simplex, real rank zero, stable rank one, and strict comparison of…

In this paper, we present algorithms for computing approximate hulls and centerpoints for collections of matrices in positive definite space. There are many applications where the data under consideration, rather than being points in a…

This article resides in the realm of the noncommutative (free) analog of real algebraic geometry - the study of polynomial inequalities and equations over the real numbers - with a focus on matrix convex sets $C$ and their projections $\hat…

In this paper we present several results on the expected complexity of a convex hull of $n$ points chosen uniformly and independently from a convex shape. (i) We show that the expected number of vertices of the convex hull of $n$ points,…

Rational maps on the Riemann sphere occupy a distinguished niche in the general theory of smooth dynamical systems. First, rational maps are complex-analytic, so a broad spectrum of techniques can contribute to their study (quasiconformal…

In this paper we present short algebraic proofs of the Linear Conway--Gordon--Sachs and the Linear van Kampen--Flores theorems in the spirit of the Radon theorem on convex hulls. {\bf Theorem.} {\it Take any $n+3$ general position points in…