Related papers: Minkowski gauges and deviation measures
We consider probability measures on $A^N$, the set of sequences of symbols on a finite alphabet $A$ of length $N$, that give a weight to each sequence in terms of a collection of matrices with non-negative entries and having rows and…
In this paper, we generalize Minkowski's theorem. This theorem is usually stated for a centrally symmetric convex body and a lattice both included in $\mathbb{R}^n$. In some situations, one may replace the lattice by a more general set for…
We construct and parametrize solutions to the constraint equations of general relativity in a neighborhood of Minkowski spacetime with arbitrary prescribed decay properties at infinity. We thus provide a large class of initial data for the…
We generalize the measurement using an expanded concept of cover, in order to provide a new approach to size of set other than cardinality. The generalized measurement has application backgrounds such as a generalized problem in dimension…
In Minkowski geometry the metric features are based on a compact convex body containing the origin in its interior. This body works as a unit ball with its boundary formed by the unit vectors. Using one-homogeneous extension we have a…
Using an optimal containment approach, we quantify the asymmetry of convex bodies in $\mathbb{R}^n$ with respect to reflections across affine subspaces of a given dimension. We prove general inequalities relating these ''Minkowski…
We derive the stability result of the dual curvature measure with near constant density in the even case. As an application, the existence and uniqueness of solutions to the even dual Minkowski problem for positive indices in…
It is easy to show that the lower and the upper box dimensions of a bounded set in Euclidean space are invariant with respect to the ambient space. In this article we show that the Minkowski content of a Minkowski measurable set is also…
Existing theories on deep nonparametric regression have shown that when the input data lie on a low-dimensional manifold, deep neural networks can adapt to the intrinsic data structures. In real world applications, such an assumption of…
In this paper, we study the conjecture of Gardner and Zvavitch from \cite{GZ}, which suggests that the standard Gaussian measure $\gamma$ enjoys $\frac{1}{n}$-concavity with respect to the Minkowski addition of \textbf{symmetric} convex…
The general expression of the angular distance between two point sources as measured by an arbitrary observer is given. The modelling presented here is rigorous, covariant and valid in any space-time. The sources of light may be located at…
In this paper, we study properties of certain risk measures associated with acceptance sets. These sets describe regulatory preconditions that have to be fulfilled by financial institutions to pass a given acceptance test. If the financial…
Several measures of non-convexity (departures from convexity) have been introduced in the literature, both for sets and functions. Some of them are of geometric nature, while others are more of topological nature. We address the statistical…
In the first part of the paper, we define an approximated Brunn-Minkowski inequality which generalizes the classical one for length spaces. Our new definition based only on distance properties allows us also to deal with discrete spaces.…
In the present paper, we study a set that can be treated as a generalised set of subsums for a geometric series. This object was discovered independently in various mathematical aspects. For instance, it is closely related to various…
We consider a different $L^p$-Minkowski combination of compact sets in $\mathbb{R}^n$ than the one introduced by Firey and we prove an $L^p$-Brunn-Minkowski inequality, $p \in [0,1]$, for a general class of measures called convex measures…
Symmetry is one of the most general and useful concepts in physics. A theory or a system that has a symmetry is fundamentally constrained by it. The same constraints do not apply when the symmetry is broken. The quantitative determination…
The class of convex sets that admit approximations as Minkowski sum of a compact convex set and a closed convex cone in the Hausdorff distance is introduced. These sets are called approximately Motzkin-decomposable and generalize the notion…
In this paper a measure of non-convexity for a simple polygonal region in the plane is introduced. It is proved that for "not far from convex" regions this measure does not decrease under the Minkowski sum operation, and guarantees that the…
In this note we prove that on general metric measure spaces the perimeter is equal to the relaxation of the Minkowski content w.r.t.\ convergence in measure