Related papers: Isoperimetric problems in sectors with density

The thesis concentrates on two problems in discrete geometry, whose solutions are obtained by analytic, probabilistic and combinatoric tools. The first chapter deals with the strong polarization problem. This states that for any sequence…

We give upper bounds for the density of unit ball packings relative to their outer parallel domains and discuss their connection to contact numbers. Also, packings of soft balls are introduced and upper bounds are given for the fraction of…

We solve a class of isoperimetric problems on $\mathbb{R}^N $ with respect to weights that are powers of the distance to the origin. For instance we show that if $k\in [0,1]$, then among all smooth sets $\Omega$ in $\mathbb{R} ^N$ with…

We discuss the smoothness and strict convexity of the solution of the $L_p$ Minkowski problem when $p<1$ and the given measure has a positive density function.

We have previously explored cylindrical packings of disks and their relation to sphere packings. Here we extend the analytical treatment of disk packings, analysing the rules for phyllotactic indices of related structures and the variation…

We prove a quantitative version of the isoperimetric inequality for a non local perimeter of Minkowski type. We also apply this result to study isoperimetric problems with repulsive interaction terms, under convexity constraints. We show…

In this paper, we prove some isoperimetric inequalities and give a sharp bound for the positive solution of sublinear elliptic equations.

A non-iterative method is presented for the factorization step of sector decomposition method, which separates infrared divergent part from loop integration. This method is based on a classification of asymptotic behavior of polynomials.…

We prove a relative isoperimetric inequalities for Lagrangian half disks in $\mathbb{C}^2$ with respect to a Lagrangian plane, or a complex plane, or a union of any two of Lagrangian or complex planes that intersect transversally at the…

We consider the problem of testing, for a given set of planar regions $\cal R$ and an integer $k$, whether there exists a convex shape whose boundary intersects at least $k$ regions of $\cal R$. We provide a polynomial time algorithm for…

The paper [1] considers a spherical cloak described by three radially varying acoustical quantities. For a given radial mass density in the cloak the question posed is whether the remaining two parameters, tangential density and…

This paper is devoted to Riemann-Hilbert problems with constraints. We obtain results characterizing the existence of solutions as well as the dimension of the solution space in terms of certain indices. As an application, we show how such…

Locally isoperimetric $N$-partitions are partitions of the space $\mathbb R^d$ into $N$ regions with prescribed, finite or infinite measure, which have minimal perimeter (which is the $(d-1)$-dimensional measure of the interfaces between…

We prove some isoperimetric type inequalities for real harmonic functions in the unit disk belonging to the Hardy space $h^p$, $p>1$ and for complex harmonic functions in $h^4$. The results extend some recent results on the area. Further we…

It has been known for almost 200 years that some angles cannot be trisected by straightedge and compass alone. This paper studies the set of such angles as well as its complement $\mathcal{T}$, both regarded as subsets of the unit circle…

Electromagnetic Inverse Scattering Problems (EISP) have gained wide applications in computational imaging. By solving EISP, the internal relative permittivity of the scatterer can be non-invasively determined based on the scattered…

The geometry of closed surfaces equipped with a Euclidean metric with finitely many conical points of arbitrary angle is studied. The main result is that the set of closed geodesics is dense in the space of geodesics.

We investigate the properties of a dark matter sector where supersymmetry is a good symmetry. In this context we find that the stability of the dark matter candidate is possible even when R-parity is broken in the visible sector. In order…

We consider the integro-differential equation ${\rm I}^{\alpha}_{0+}f= x^m f$ on the half-line. We show that there exists a density solution, which is then unique and can be expressed in terms of the Beta distribution, if and only if $m>…

The authors propose a Nystrom method to approximate the solution of a boundary integral equation connected with the exterior Neumann problem for Laplace's equation on planar domains with corners. They prove the convergence and the stability…