Minkowski compactness measure

This is the author accepted manuscript. The final version is available from the publisher via the DOI in this record.Published in: Computational Intelligence (UKCI), 2013, 13th UK Workshop, Guildford UK. Date of Conference: 9-11 Sept. 2013Many compactness measures are available in the literature. In this paper we present a generalised compactness measure Cq(S) which unifies previously existing definitions of compactness. The new measure is based on Minkowski distances and incorporates a parameter q which modifies the behaviour of the compactness measure. Different shapes are considered to be most compact depending on the value of q: for q = 2, the most compact shape in 2D (3D) is a circle (a sphere); for q → ∞, the most compact shape is a square (a cube); and for q = 1, the most compact shape is a square (a octahedron). For a given shape S, measure Cq(S) can be understood as a function of q and as such it is possible to calculate a spectum of Cq(S) for a range of q. This produces a particular compactness signature for the shape S, which provides additional shape information. The experiments section of this paper provides illustrative examples where measure Cq(S) is applied to various shapes and describes how measure and its spectrum can be used for image processing applications

Minimizers for nonlocal perimeters of Minkowski type

We study a nonlocal perimeter functional inspired by the Minkowski content, whose main feature is that it interpolates between the classical perimeter and the volume functional. This problem is related by a generalized coarea formula to a Dirichlet energy functional in which the energy density is the local oscillation of a function. These two nonlocal functionals arise in concrete applications, since the nonlocal character of the problems and the different behaviors of the energy at different scales allow the preservation of details and irregularities of the image in the process of removing white noises, thus improving the quality of the image without losing relevant features. In this paper, we provide a series of results concerning existence, rigidity and classification of minimizers, compactness results, isoperimetric inequalities, Poincar\'e-Wirtinger inequalities and density estimates. Furthermore, we provide the construction of planelike minimizers for this generalized perimeter under a small and periodic volume perturbation.Comment: To appear in Calc. Var. Partial Differential Equation

A Remark on the Anisotropic Outer Minkowski content

We study an anisotropic version of the outer Minkowski content of a closed set in Rn. In particular, we show that it exists on the same class of sets for which the classical outer Minkowski content coincides with the Hausdorff measure, and we give its explicit form.Comment: We corrected an error in the orignal manuscript, on p. 14 (the boundaries of the regularized sets are not necessarily C^{1,1}

Improved sphere packing lower bounds from Hurwitz lattices

In this paper we prove an asymptotic lower bound for the sphere packing density in dimensions divisible by four. This asymptotic lower bound improves on previous asymptotic bounds by a constant factor and improves not just lower bounds for the sphere packing density, but also for the lattice sphere packing density and, in fact, the Hurwitz lattice sphere packing density.Comment: 12 page

Discreteness without symmetry breaking: a theorem

This paper concerns sprinklings into Minkowski space (Poisson processes). It proves that there exists no equivariant measurable map from sprinklings to spacetime directions (even locally). Therefore, if a discrete structure is associated to a sprinkling in an intrinsic manner, then the structure will not pick out a preferred frame, locally or globally. This implies that the discreteness of a sprinkled causal set will not give rise to ``Lorentz breaking'' effects like modified dispersion relations. Another consequence is that there is no way to associate a finite-valency graph to a sprinkling consistently with Lorentz invariance.Comment: 7 pages, laTe

Isoperimetric problems for a nonlocal perimeter of Minkowski type

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 existence of minimizers, and we describe the shape of minimizers in certain parameter regimes