Related papers: On upper conical density results
We study continuity and discontinuity properties of some popular measure-dimension mappings under some topologies on the space of probability measures in this work. We give examples to show that no continuity can be guaranteed under general…
The purpose of this paper is twofold. First, we describe one (presumably) new case, in which Busemann--Hausdorff densities are convex. We apply the corresponding result to prove the existence of minimizing rectifiable chains of codimension…
In this paper we prove an upper bound on the "size" of the set of multiplicatively $\psi$-approximable points in $\mathbb R^d$ for $d>1$ in terms of $f$-dimensional Hausdorff measure. This upper bound exactly complements the known lower…
The Hausdorff dimension of the set of points that are covered infinitely many times by a sequence of randomly distributed balls in the unit cube can be expressed in terms of the sizes of the balls. This note presents a new proof of the…
We prove uniform Hausdorff and packing dimension results for the inverse images of a large class of real-valued symmetric L\'evy processes. Our main result for the Hausdorff dimension extends that of Kaufman (1985) for Brownian motion and…
In this paper we prove a theorem that provides an upper bound for the density of packings of congruent copies of a given convex body in $\mathbb{R}^n$; this theorem is a generalization of the linear programming bound for sphere packings. We…
The aim of this paper is to study ultralimits of pointed metric measure spaces (possibly unbounded and having infinite mass). We prove that ultralimits exist under mild assumptions and are consistent with the pointed measured…
The aim of this paper is to extend the concept of measure density introduced by Buck for finite unions of arithmetic progressions, to arbitrary subsets of N defined by a given system of decompositions. This leads to a variety of new…
We deal with finitely additive measures defined on all subsets of natural numbers which extend the asymptotic density (density measures). We consider a class of density measures which are constructed from free ultrafilters on natural…
The Kepler conjecture asserts that no packing of congruent balls in three-dimensional Euclidean space has density greater than that of the face-centered cubic packing. The original proof, announced in 1998 and published in 2006, is long and…
The conic support measures localize the conic intrinsic volumes of closed convex cones in the same way as the support measures of convex bodies localize the intrinsic volumes of convex bodies. In this note, we extend the `Master Steiner…
We prove the existence of isoperimetric regions in $\R^n$ with density under various hypotheses on the growth of the density. Along the way we prove results on the boundedness of isoperimetric regions.
The paper treats density measures as typical examples of finitely additive measures in $\mathbb{R}^n$. We study their structure and derive basic properties. In addition, estimates for related integrals are provided. The results are applied…
We prove a Tb theorem on quasimetric spaces equipped with what we call an upper doubling measure. This is a property that encompasses both the doubling measures and those satisfying the upper power bound \mu(B(x,r)) \le Cr^d. Our spaces are…
We study here a sequence of secondary measures, so called because the set of secondary polynomials on a given term become orthogonal for the next measure. The main result is a formula making explicit the density of any term of the sequence,…
In this paper, we study the multifractal Hausdorff and packing dimensions of Borel probability measures and study their behaviors under orthogonal projections. In particular, we try through these results to improve the main result of M. Dai…
Let E be a locally compact second countable Hausdorff space and F the pertaining family of all closed sets. We endow F respectively with the Fell-topology, the upper Fell topology or the upper Vietoris-topology and investigate weak…
The Kepler conjecture asserts that no packing of congruent balls in three-dimensional Euclidean space has density greater than that of the face-centered cubic packing. In 1998, Sam Ferguson and I announced a computer-assisted proof of this…
A recently proposed convolution technique for the calculation of local density of states is described more thouroughly and new results of its application are presented. For separable systems the exposed method allows to construct the ldos…
We investigate the influence that $s$-dimensional lower and upper Hausdorff densities have on the geometry of a Radon measure in $\mathbb{R}^n$ when $s$ is a real number between $0$ and $n$. This topic in geometric measure theory has been…