Related papers: Geometric inequalities, stability results and Kend…
Kendall's Shape Theory covers shapes formed by $N$ points in $\mathbb{R}^d$ upon quotienting out the similarity transformations. This theory is based on the geometry and topology of the corresponding configuration space: shape space.…
Investigating a model of scale-invariant random spatial network suggested by Aldous, Kendall constructed a random metric $T$ on $\mathbb{R}^d$, for which the distance between points is given by the optimal connection time, when travelling…
We establish higher-order nonasymptotic expansions for a difference between probability distributions of sums of i.i.d. random vectors in a Euclidean space. The derived bounds are uniform over two classes of sets: the set of all Euclidean…
In this paper, we study Wicksell's corpuscle problem in spaces of constant curvature, thus extending the classical Euclidean framework. We consider a particle process of balls with random radii in such a space, assumed to be invariant under…
Performance accuracy of the Euclidean Distance Discriminant rule (EDDR) is studied in the high-dimensional asymptotic framework which allows the dimensionality to exceed sample size. Under mild assumptions on the traces of the covariance…
It has been proposed that the dynamical stability of topological defects in the visual cortex reflects the Euclidean symmetry of the visual world. We analyze defect stability and pattern selection in a generalized Swift-Hohenberg model of…
The union of the particles of a stationary Poisson process of compact (convex) sets in Euclidean space is called Boolean model and is a classical topic of stochastic geometry. In this paper, Boolean models in hyperbolic space are…
We study the asymptotic distribution, as the volume parameter goes to 1, of the peak (largest part) of finite- or slowly-growing-width cylindric plane partitions weighted by their trace, seam, and volume. There are two natural asymptotic…
The appealing connection between non-Euclidean geometries and defects in solids is brought forth in this article. Drawing a correspondence between the nature of a defect and a specific geometric property of the material space not only…
We consider several geometric inequalities in general relativity involving mass, area, charge, and angular momentum for asymptotically hyperboloidal initial data. We show how to reduce each one to the known maximal (or time symmetric) case…
This article belongs to the area of geometric tomography, which is the study of geometric properties of solids based on data about their sections and projections. We describe a new direction in geometric tomography where different…
The asymptotic expansion method is generalized from the periodic setting to stationary ergodic stochastic geometries. This will demonstrate that results from periodic asymptotic expansion also apply to non-periodic structures of a certain…
Geometrical model for material Dirac wave field and for Maxwell electromagnetic field is suggested where above fields are considered as propagating regions of the space itself with distorted euclidean geometry. It is shown that equations…
We consider a shape optimization problem related to the persistence threshold for a biological species, the unknown shape corresponding to the zone of the habitat which is favorable to the population. Analytically, this translates in the…
We observe a realization of a stationary generalized weighted Voronoi tessellation of the d-dimensional Euclidean space within a bounded observation window. Given a geometric characteristic of the typical cell, we use the minus-sampling…
Motivated by Felix Klein's notion that geometry is governed by its group of symmetry transformations, Charles Ehresmann initiated the study of geometric structures on topological spaces locally modeled on a homogeneous space of a Lie group.…
Uniformity and proximity are two different ways for defining small scale structures on a set. Coarse structures are large scale counterparts of uniform structures. In this paper, motivated by the definition of proximity, we develop the…
Magnitude is an isometric invariant of metric spaces inspired by category theory. Recent work has shown that the asymptotic behavior under rescaling of the magnitude of subsets of Euclidean space is closely related to intrinsic volumes.…
This is a chapter of a forthcoming Lecture Notes in Mathematics "Modern Approaches to Discrete Curvature" edited by L. Najman and P. Romon. It provides a survey on geometric and spectral consequences of curvature bounds. The geometric…
Transverse wrinkles are known to appear in thin rectangular elastic sheets when stretched in the long direction. Numerically computed bifurcation diagrams for extremely thin, highly stretched films indicate entire orbits of wrinkling…