Related papers: Polynomials associated to non-convex bodies

We associate convex regions in R^n to m-primary graded sequences of subspaces, in particular m-primary graded sequences of ideals, in a large class of local algebras (including analytically irreducible local domains). These convex regions…

Orthogonal polynomials have very useful properties in the solution of mathematical problems, so recent years have seen a great deal in the field of approximation theory using orthogonal polynomials. In this paper, we characterize the…

We consider here a particular quadratic equation linking two elements of a C-Algebra. By analysing powers of the unknowns, it appears a double sequence of polynomials related to classical Bernoulli polynomials. We get the generating…

In this note we show that unbounded convex polygons with nonparallel unbounded edges are polynomial images of ${\mathbb R}^2$, whereas their interiors are polynomial images of ${\mathbb R}^3$

This article is concerned with the approximation of unbounded convex sets by polyhedra. While there is an abundance of literature investigating this task for compact sets, results on the unbounded case are scarce. We first point out the…

We consider a stationary Poisson hyperplane process with given directional distribution and intensity in $d$-dimensional Euclidean space. Generalizing the zero cell of such a process, we fix a convex body $K$ and consider the intersection…

Let $K \subset \R^d$ be a smooth convex set and let $\P_\la$ be a Poisson point process on $\R^d$ of intensity $\la$. The convex hull of $\P_\la \cap K$ is a random convex polytope $K_\la$. As $\la \to \infty$, we show that the variance of…

In this paper, we give an overview of some results concerning best and random approximation of convex bodies by polytopes. We explain how both are linked and see that random approximation is almost as good as best approximation.

The distance between convex bodies \(K, L \subseteq \R^n\) is defined as \[ d(K,L)= \inf \left\{ \lambda \ge 1: \ L-x \subseteq T (K-y) \subseteq \lambda (L-x) \right\}, \] where the infimum is taken over all \(x,y \in \R^n\) and all…

If the complement of a closed convex set in a closed convex cone is bounded, then this complement minus the apex of the cone is called a coconvex set. Coconvex sets appear in singularity theory (they are closely related to Newton diagrams)…

We consider properties and applications of a sequence of polynomials known as complementary Romanovski-Routh polynomials (CRR polynomials for short). These polynomials, which follow from the Romanovski-Routh polynomials or complexified…

In the paper "Isoperimetry of waists and local versus global asymptotic convex geometries", it was proved that the existence of nicely bounded sections of two symmetric convex bodies K and L implies that the intersection of randomly rotated…

The goal of the present paper is to introduce and study noncommutative Hardy spaces associated with the regular $\Lambda$-polyball, to develop a functional calculus on noncommutative Hardy spaces for the completely non-coisometric (c.n.c.)…

We study properties of convex hulls of (co)adjoint orbits of compact groups, with applications to invariant theory and tensor product decompositions. The notion of partial convex hulls is introduced and applied to define two numerical…

Extending classical results on polytopal approximation of convex bodies, we derive asymptotic formulas for the weighted approximation of smooth convex functions by piecewise affine convex functions as the number of their facets tends to…

We associate convex bodies to a wide class of graded G-algebras where G is a connected reductive group. These convex bodies give information about the Hilbert function as well as multiplicities of irreducible representations appearing in…

We study polyhedral approximations to the cone of nonnegative polynomials. We show that any constant ratio polyhedral approximation to the cone of nonnegative degree $2d$ forms in $n$ variables has to have exponentially many facets in terms…

We prove that for every convex body $K$ with the center of mass at the origin and every $\varepsilon\in \left(0,\frac{1}{2}\right)$, there exists a convex polytope $P$ with at most $e^{O(d)}\varepsilon^{-\frac{d-1}{2}}$ vertices such that…

We give explicit polynomial-sized (in $n$ and $k$) semidefinite representations of the hyperbolicity cones associated with the elementary symmetric polynomials of degree $k$ in $n$ variables. These convex cones form a family of…

We describe the asymptotic behavior of the multivariate BC-type Jacobi polynomials as the number of variables and the Young diagram indexing the polynomial go to infinity. In particular, our results describe the approximation of the…