Related papers: Optimal polynomial meshes exist on any multivariat…

In this paper, we prove several theorems on systems of polynomials with at least one positive real zero based on the theory of conceive polynomials. These theorems provide sufficient conditions for systems of multivariate polynomials…

We present an algorithm that enumerates and classifies all edge-to-edge gluings of unit squares that correspond to convex polyhedra. We show that the number of such gluings of $n$ squares is polynomial in $n$, and the algorithm runs in time…

In this work we prove that, for a general polyhedral domain of $\mathbb{R}^3$, the cohomology spaces of the discrete de Rham complex of [Di Pietro and Droniou, An arbitrary-order discrete de Rham complex on polyhedral meshes: Exactness,…

We consider the following geometric optimization problem: find a maximum-area rectangle and a maximum-perimeter rectangle contained in a given convex polygon with $n$ vertices. We give exact algorithms that solve these problems in time…

Let $K$ be a compact subset of a totally-real manifold $M$, where $M$ is either a $\mathcal{C}^2$-smooth graph in $\mathbb{C}^{2n}$ over $\mathbb{C}^n$, or $M=u^{-1}\{0\}$ for a $\mathcal{C}^2$-smooth submersion $u$ from $\mathbb{C}^n$ to…

Suppose that $K \subseteq \RR^d$ is a 0-symmetric convex body which defines the usual norm $$ \Norm{x}_K = \sup\Set{t\ge 0: x \notin tK} $$ on $\RR^d$. Let also $A\subseteq\RR^d$ be a measurable set of positive upper density $\rho$. We show…

Let $M$ be a strictly convex smooth connected hypersurface in $\mathbb R^n$ and $\widehat{M}$ its convex hull. We say that $M$ is locally polynomially integrable if the $(n-1)-$ dimensional volumes of the sections of $\widehat M$ by…

We design in this work a discrete de Rham complex on manifolds. This complex, written in the framework of exterior calculus, has the same cohomology as the continuous de Rham complex, is of arbitrary order of accuracy and, in principle, can…

The X-ray numbers of some classes of convex bodies are investigated. In particular, we give a proof of the X-ray Conjecture as well as of the Illumination Conjecture for almost smooth convex bodies of any dimension and for convex bodies of…

In this paper we prove the conjecture of Korevaar and Meyers: for each $N\ge c_dt^d$ there exists a spherical $t$-design in the sphere $S^d$ consisting of $N$ points, where $c_d$ is a constant depending only on $d$.

The generalized Lax conjecture asserts that each hyperbolicity cone is a linear slice of the cone of positive semidefinite matrices. We prove the conjecture for a multivariate generalization of the matching polynomial. This is further…

This article investigates the numerical approximation of shape optimization problems with PDE constraint on classes of convex domains. The convexity constraint provides a compactness property which implies well posedness of the problem.…

We give a short proof of a result obtained by Mohammad Ghomi concerning existence of nets of a convex polyhedron after a suitable linear transformation.

We show that minimizing a convex function over the integer points of a bounded convex set is polynomial in fixed dimension.

We give a direct global proof for the existence of symplectic realizations of arbitrary Poisson manifolds.

The Generalized Lax Conjecture asks whether every hyperbolicity cone is a section of a semidefinite cone of sufficiently high dimension. We prove that the space of hyperbolicity cones of hyperbolic polynomials of degree $d$ in $n$ variables…

For a given $\lambda >0$, a convex body in $\mathbb R^n$ is $\lambda$-convex if it is the intersection of (finitely or infinitely many) balls of radius $1/\lambda$. In this note, we show that among all $\lambda$-convex bodies in $\mathbb…

Similarly to the classic notion in $E^d$, a subset of a positive diameter below $\frac{\pi}{2}$ of a hemisphere of the sphere $S^d$ is called complete, provided adding any extra point increases its diameter. Complete sets are convex bodies…

In this paper we prove a recent conjecture formulated by Dmitrishin, Smorodin and Stokolos about that certain polynomials are univalent in the unit disk. As a consequence we get an upper estimate for the Koebe radius of univalent…

We study the location and the size of the roots of Steiner polynomials of convex bodies in the Minkowski relative geometry. Based on a problem of Teissier on the intersection numbers of Cartier divisors of compact algebraic varieties it was…