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In this paper we discuss about properties of lattices and its application in theoretical and algorithmic number theory. This result of Minkowski regarding the lattices initiated the subject of Geometry of Numbers, which uses geometry to study the properties of algebraic numbers. It has application on various other fields of mathematics especially the study of Diophantine equations, analysis of functional analysis etc. This paper will review all the major developments that have occurred in the field of geometry of numbers. In this paper we shall first give a broad overview of the concept of lattice and then discuss about the geometrical properties it has and its applications., Comment: 14 Pages,36 References

We use the definition of a fractional integral, recently proposed by Katugampola, to establisha generalization of the reverse Minkowski’s inequality. We show two new theorems associatedwith this inequality, as well as state and show other inequalities related to this fractional operator.

This note geometrically generalizes the main theorem in Shahgholian’s 1992 paper [2] , from R 3 to R n ≥ 2 via the Minkowski functional of a convex body with the origin in its interior.

We prove the existence of classical positive radial solutions to the boundary value problems $$\begin{aligned} \left\{ \begin{array}{ll} -\,\text {div}\big (\frac{\nabla y}{\sqrt{1-|\nabla y|^2}}\big )=\lambda a(|x|)f(y) \ \ \ \ \ &{} \text {in}\ B(b),\\ y=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \; &{} \text {on}\ \partial B(b),\\ \end{array} \right\} . \end{aligned}$$ where $$b>0$$ , $$B(b)=\{x\in \mathbb {R}^N:|x|0$$ in [0, b], and $$\lambda >0$$ is sufficiently small. Our approach is based on the Leray–Schauder fixed point theorem.

We point out a mistake in the proof of the main theorem in a recent article on a family of generalized Minkowski's question-mark functions, saying that each member of the family is a singular homeomorphism, and provide two alternative proofs, one based on the ergodicity of the Gauss map G and the α -Luroth map L α , and another one focusing more on classical properties of continued fraction expansions.

In this paper we extend the concepts of L p -mixed volumes and L p -surface area measures to L p -mixed μ-measures and L p -surface μ-area measures, respectively, for a measure μ on R n . We give an integral representation and discuss some inequalities for the L p -surface μ-area measure. The L p -Minkowski and L p -Brunn–Minkowski inequalities for measures are proved. Moreover, we also show the existence theorem of the related L p -Minkowski problem for measures.

This note presents a proof of the Farkas–Minkowski theorem. Our proof does not presuppose the closedness of a finitely generated cone, nor employs separation theorems either. Even the concept of linear independence or invertibility of matrices is not necessary. Our new device consists in proving the Farkas–Minkowski theorem and the closedness of a finitely generated cone at the same time based upon mathematical induction. We make use of a minimization problem with an equality constraint, a method familiar to economics students.

We propose a method for computing the Minkowski sum of two free-form surfaces, given in a tensor product B-spline representation in R 3 . The Minkowski sum (typically a three dimensional volume), is represented by its boundary surface(s), also referred to as the envelope. The envelope is obtained via an algebraic equation solving approach (in the parameter space of the input geometries), followed by mapping the parametric solution to Euclidean space and filtering of redundant solution patches. The suggested method is applicable to a fairly general class of input surfaces, allowing non-convex regions, boundary curves and C 1 discontinuities, while providing a solution with topological guarantee. Test results are provided, demonstrating the suggested method using a triangular mesh approximation of the Minkowski sum envelope surface.

The aim of this paper is to generalize Minkowski’s theorem. This theorem is usually stated for a centrally symmetric convex body and a lattice both included in $$\mathbf {R}^n$$ . In some situations, one may replace the lattice by a more general set for which a notion of density exists. In this paper, we prove a Minkowski theorem for quasicrystals, which bounds from below the frequency of differences appearing in the quasicrystal and belonging to a centrally symmetric convex body. The last part of the paper is devoted to quite natural applications of this theorem to Diophantine approximation and to discretization of linear maps.

In this paper, we use the critical point theory for convex, lower semicontinuous perturbations of C 1 {C^{1}} -functionals to obtain the existence of multiple nontrivial solutions for one parameter potential systems involving the operator u ↦ div ⁡ ( ∇ ⁡ u 1 - | ∇ ⁡ u | 2 ) {u\mapsto\operatorname{div}(\frac{\nabla u}{\sqrt{1-|\nabla u|^{2}}})} . The solvability of a general non-potential system is also established.

In this paper we investigate the existence of rotationally symmetric entire graphs (resp. entire spacelike graphs) with prescribed k-th mean curvature function in Euclidean space R n + 1 (resp. Minkowski spacetime L n + 1 ). As a previous step, we analyze the associated homogeneous Dirichlet problem on a ball, which is not elliptic for k > 1 , and then we prove that it is possible to extend the solutions. Moreover, a sufficient condition for uniqueness is given in both cases.

Recently, the first author has extended the definition of the zeta function associated with fractal strings to arbitrary bounded subsets $A$ of the $N$-dimensional Euclidean space ${\mathbb R}^N$, for any integer $N\ge1$. It is defined by $\zeta_A(s)=\int_{A_{\delta}}d(x,A)^{s-N}\,\mathrm{d} x$ for all $s\in\mathbb{C}$ with $\operatorname{Re}\,s$ sufficiently large, and we call it the distance zeta function of $A$. Here, $d(x,A)$ denotes the Euclidean distance from $x$ to $A$ and $A_{\delta}$ is the $\delta$-neighborhood of $A$, where $\delta$ is a fixed positive real number. We prove that the abscissa of absolute convergence of $\zeta_A$ is equal to $\overline\dim_BA$, the upper box (or Minkowski) dimension of $A$. Particular attention is payed to the principal complex dimensions of $A$, defined as the set of poles of $\zeta_A$ located on the critical line $\{\mathop{\mathrm{Re}} s=\overline\dim_BA\}$, provided $\zeta_A$ possesses a meromorphic extension to a neighborhood of the critical line. We also introduce a new, closely related zeta function, $\tilde\zeta_A(s)=\int_0^{\delta} t^{s-N-1}|A_t|\,\mathrm{d} t$, called the tube zeta function of $A$. Assuming that $A$ is Minkowski measurable, we show that, under some mild conditions, the residue of $\tilde\zeta_A$ computed at $D=\dim_BA$ (the box dimension of $A$), is equal to the Minkowski content of $A$. More generally, without assuming that $A$ is Minkowski measurable, we show that the residue is squeezed between the lower and upper Minkowski contents of $A$. We also introduce transcendentally quasiperiodic sets, and construct a class of such sets, using generalized Cantor sets, along with Baker's theorem from the theory of transcendental numbers., Comment: 54 pages, corrected misprints, reduced number of self-citations

In this paper we provide several characterizations of Minkowski sets, i.e. closed, possibly unbounded, convex sets which are representable as the convex hulls of their sets of extreme points. The equality between the relative boundary of a closed convex set containing no lines and its Pareto-like associated set ensures the Minkowski property of the set. In two dimensions this equality characterizes the Minkowski sets containing no lines.

Este trabalho tem por objetivo iniciar o estudo na Geometria Aritmética com o foco no célebre Teorema do Minkowski sobre reticulados. O intuito é fazer um estudo abrangente desde a definição de reticulados, até as aplicações deste teorema em Teoria Elementar dos Números, Aproximação Diofantina e Teoria Algébrica dos Números. Para esse propósito, algumas definições e conceitos básicos são dados, mas abordados de forma sucinta. The aim of this work is to be an early study on Arithmetic Geometry, focused on the famous Minkowski's Theorem on lattices. Our goal is to make a substantial study from the definition of lattices until some applications of this main theorem on well-known results on Elementary Number Theopry, Diophantine Approximation and Algebraic Number Theory. For this purpose, some basic definitions and concepts are given, but approached in a brief way. Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq

The Dirichlet–Voronoi cell and parallelohedron are fundamental concepts in Geometry. In particular, they do play important roles in the study of ball packing and ball covering. However, to study packing and covering of general convex bodies, they are no longer so useful (see Theorem 0). By introducing Minkowski bisectors and Minkowski cells, this paper explores a new way to study the density $$\theta ^*(C)$$ of the thinnest lattice covering of $$\mathbb {E}^n$$ by a centrally symmetric convex body C. Several basic results (Theorems 2 and 4, Corollary 1) and unexpected geometric phenomena (Theorem 0, Example 1, Remark 4) about Minkowski bisectors, Minkowski cells and covering densities are discovered.

The Minkowski length of a lattice polytope $P$ is a natural generalization of the lattice diameter of $P$. It can be defined as the largest number of lattice segments whose Minkowski sum is contained in $P$. The famous Ehrhart theorem states that the number of lattice points in the positive integer dilates $tP$ of a lattice polytope $P$ behaves polynomially in $t\in\mathbb{N}$. In this paper we prove that for any lattice polytope $P$, the Minkowski length of $tP$ for $t\in\mathbb{N}$ is eventually a quasi-polynomial with linear constituents. We also give a formula for the Minkowski length of coordinates boxes, degree one polytopes, and dilates of unimodular simplices. In addition, we give a new bound for the Minkowski length of lattice polygons and show that the Minkowski length of a lattice triangle coincides with its lattice diameter., Comment: 13 pages, 1 figure; minor corrections, to appear in European Journal of Combinatorics

We study the Minkowski symmetry set of a closed smooth curveγin the Minkowski plane. We answer the following question, which is analogous to one concerning curves in the Euclidean plane that was treated by Giblin and O’Shea (1990): given a pointponγ, does there exist a bi-tangent pseudo-circle that is tangent toγboth atpand at some other pointqonγ? The answer is yes, but as pseudo-circles with non-zero radii have two branches (connected components) it is possible to refine the above question to the following one: given a pointponγ, does there exist a branch of a pseudo-circle that is tangent toγboth atpand at some other pointqonγ? This question is motivated by the earlier quest of Reeve and Tari (2014) to define the Minkowski Blum medial axis, a counterpart of the Blum medial axis of curves in the Euclidean plane.

Monica Klein classified Minkowski planes with respect to subgroups of Minkowski homotheties. In this paper we investigate finite Minkowski planes with respect to groups of automorphism of Klein type 20. We show that this type can only occur in miquelian planes.

We validate the conjectured logarithmic Minkowski inequality, and thus the equivalent logarithmic Brunn-Minkowski inequality, in some particular cases and we prove some variants of the logarithmic Minkowski inequality for general convex bodies without the symmetry assumption. An application of one of these variants is shown.

We introduce and study a sequence of geometric invariants for convex bodies in finite-dimensional spaces, which is in a sense dual to the sequence of mean Minkowski measures of symmetry proposed by the second author. It turns out that the sequence introduced in this paper shares many nice properties with the sequence of mean Minkowski measures, such as the sub-arithmeticity and the upper-additivity. More meaningfully, it is shown that this new sequence of geometric invariants, in contrast to the sequence of mean Minkowski measures which provides information on the shapes of lower dimensional sections of a convex body, provides information on the shapes of orthogonal projections of a convex body. The relations of these new invariants to the well-known Minkowski measure of asymmetry and their further applications are discussed as well.

In this paper, we reconsider the $(1,3)$-Bertrand curves with respect to the casual characters of a $\left( 1,3\right) $-normal plane that is a plane spanned by the principal normal and the second binormal vector fields of the given curve. Here, we restrict our investigation of $(1,3)$-Bertrand curves to the spacelike $\left( 1,3\right) $-normal plane in Minkowski space-time. We obtain the necessary and sufficient conditions for the curves with spacelike $\left( 1,3\right) $-normal plane to be $(1,3)$-Bertrand curves and we give the related examples for these curves.

Necessary and sufficient conditions for the existence of the group inverse of the block matrix in Minkowski Space are studied, where are both square and . The representation of this group inverse and some related additive results are also given.

In this paper, we obtain some new characterizations of the range symmetric matrices in the Minkowski Space M by using the Block representation of the matrices. These characterizations are used to establish some results on the partial ordering of the range symmetric matrices with respect to the Minkowski adjoint. Further, we establish some results regarding the partial ordering of m-projectors with respect to the Minkowski adjoint and manipulate them to characterize some sets of range symmetric elements in the Minkowski Space M. All the results obtained in this paper are an extension to the Minkowski space of those given by A. Hernandez, et al. in [The star partial order and the eigenprojection at 0 on EP matrices, Applied Mathematics and Computation, 218: 10669-10678, 2012].

In this article, we establish some inequalities of quermassintegrals associated with mixed Blaschke Minkowski homomorphisms. In particular, Minkowski and Brunn-Minkowski type inequalities for quermassintegrals differences of mixed Blaschke Minkowski homomorphisms are established. In addition, we also give an isolated form of Brunn-Minkowski type inequality of quermassintegrals established by Schuster.

The weighted Minkowski inverse A M , N ? ? M n , m ?related to the positive definite matrices M ? M m and N ? M n of an arbitrary matrix A ? M m , n (including singular and rectangular) is one of the important generalized inverses for solving matrix equations in Minkowski space µ . In this paper, the results are introduced in the following three ways. First, we establish some new and attractive properties of the weighted Minkowski inverse A M , N ? in a Minkowski space µ . Second, new representations and conditions for the continuity of the weighted Minkowski inverse A M , N ? in a Minkowski space µ are discussed. Finally, some illustrated counterexamples are also studied to show that some well-known properties of the weighted Moore-Penrose inverse A M , N + in a Hilbert space H are not valid in the Minkowski inverse A M , N ? in a Minkowski space µ .

In this paper, we solve L p Minkowski problem for L 1 data and all p 0 , and Orlicz Minkowski problem with two nonlinear terms in L 1 sense. A byproduct is the Blaschke–Santalo inequality, which was previously established for only constant data, and now is shown to hold for L 1 data.

Hermann Minkowski introduced a function in 1904 which maps quadratic irrational numbers to rational numbers and this function is now known as Minkowski's question mark function since Minkowski used the notation $?(x)$. This function is a distribution function on $[0,1]$ which defines a singular continuous measure with support $[0,1]$. Our interest is in the (monic) orthogonal polynomials $(P_n)_{n \in \mathbb{N}}$ for the Minkowski measure and in particular in the behavior of the recurrence coefficients of the three term recurrence relation. We will give some numerical experiments using the discretized Stieltjes-Gautschi method with a discrete measure supported on the Minkowski sequence. We also explain how one can compute the moments of the Minkowski measure and compute the recurrence coefficients using the Chebyshev algorithm., Comment: 26 pages, 11 figures, 4 tables

In this article, we establish Minkowski and Aleksandrov–Fenchel type inequalities for the volume sum of polars of Blaschke–Minkowski homomorphisms.

International audience; We study the problem of reconstructing a convex body using only a finite number of measurements of outer normal vectors. More precisely, we suppose that the normal vectors are measured at independent random locations uniformly distributed along the boundary of our convex set. Given a desired Hausdorff error $\eta$, we provide an upper bounds on the number of probes that one has to perform in order to obtain an $\eta$-approximation of this convex set with high probability. Our result rely on the stability theory related to Minkowski's theorem.

Let Mn be a compact hypersurface of a Minkowski space (Vn+1, F). In this paper, using the Gauss formula of the Chern connection for Finsler submanifolds, we prove that if the second mean curvature H2 of M is constant and the norm square S of the second fundamental form of M satisfies S  n(n−1) n−2 H2, then M with the induced metric is isometric to the standard Euclidean sphere. This generalizes the result of [2] from the Euclidean to the Minkowski space.

Minkowski's second theorem in the Geometry of Numbers provides optimal upper and lower bounds for the volume of an o-symmetric convex body in terms of its successive minima. In this paper we study analogs of this theorem from two different points of view: either relaxing the symmetry condition, assuming that the centroid of the body lies at the origin, or replacing the volume functional by the surface area.

We obtain a Minkowski type inequality for convex functions with weights satisfying the Jensen-Steffensen conditions.

We consider the expansion of co-compact convex hypersurfaces in Minkowski space by functions of their curvature, and prove under quite general conditions that solutions are asymptotic to the self-similar expanding hyperboloid. In particular this implies a convergence result for a class of special solution of the cross-curvature flow of negatively curved Riemannian metrics on three-manifolds.

This paper is devoted to the study of the asymptotics of the spectrum of the boundary-value problem $$ - y'' - \lambda \rho y = 0,y(0) = y(1) = 0,$$ where ρ is the generalized derivative of the Minkowski function, i.e., ρ =?′(x) (here ?(x) is the “question-mark function” first defined by Minkowski, who introduced this notation). For the eigenvalues of the problem, asymptotic two-sided estimates of power type are obtained. The order of the power is determined by the Hausdorff dimension of the support of the Minkowski measure d?.

Motivated by applications in local stereology, a new rotational Crofton formula is derived for Minkowski tensors. For sets of positive reach, the formula shows how rotational averages of intrinsically defined Minkowski tensors on sections passing through the origin are related to the geometry of the sectioned set. In particular, for Minkowski tensors of order j − 1 on j-dimensional linear subspaces, we derive an explicit formula for the rotational average involving hypergeometric functions. Sectioning with lines and hyperplanes through the origin is considered in detail. We also study the case where the sections are not restricted to pass through the origin. For sets of positive reach, we here obtain a Crofton formula for the integral mean of intrinsically defined Minkowski tensors on j-dimensional affine subspaces.

A certain class of results about the different representations of Oblique projectors is present in the literature. These results represent Oblique projectors as the functions of orthogonal projectors with given onto and along spaces. But these results are valid under the restriction that the functions of orthogonal projectors involved are invertible. In this paper we extend and generalize these results. The extension lies in making a transition from Euclidean space to Minkowski space M and the generalization is obtain by voiding the invertibility condition and use of the Minkowski inverse. Furthermore, the nobility lies in utilizing the m-projectors instead of the regular orthogonal projectors.

We explicitly solved the anomaly-free quantum constraints proposed by Tomlin and Varadarajan for the weak Euclidean model of canonical loop quantum gravity, in a large subspace of the model's kinematic Hilbert space which is the space of the charge network states. In doing so, we first identified the subspace on which each of the constraints acts convergingly, and then by explicitly evaluating such actions we found the complete set of the solutions in the identified subspace. We showed that the space of solutions consists of two classes of states, with the first class having a property that involves the condition known from the Minkowski theorem on polyhedra, and the second class satisfying a weaker form of the spatial diffeomorphism invariance.

The HOL Light proof assistant is famous for its huge multivariate analysis library: nearly 300,000 lines of code and 13,000 theorems. A substantial fraction of this library has been manually ported to Isabelle/HOL. The Isabelle analysis library contains approximately 7400 named theorems, including Cauchy's integral and residue theorems, the Liouville theorem, the open mapping and domain invariance theorems, the maximum modulus principle and the Krein-Milman Minkowski theorem. Why port proofs manually given so much work on porting proofs automatically? Typical approaches rely on low level encodings that seldom yield natural-looking results. Manual porting has allowed us to generalise many results from n-dimensional vector spaces to metric or topological spaces. The transition from the traditional LCF/HOL proof style (which has hardly changed since 1985) to structured proofs has produced a dramatic improvement in the legibility of the material. Automatic porting generally yields a list of theorem statements but no intelligible proofs. This project has highlighted three features of Isabelle working well together: heuristic automation, structured proofs and sledgehammer. Heuristic automation builds in a lot of implicit knowledge, which is potentially unpredictable, but in combination with structured proofs any breakages (caused by updates to the system) are localised and easily fixed. Sledgehammer (which uses powerful external automation to solve subgoals) can frequently complete an argument without requiring a precise reproduction of the original HOL Light proof. Sledgehammer also encourages a style in which the user reaches the desired result by suggesting a series of intermediate claims. Such proofs are genuinely human-oriented. And only such proofs will attract mathematicians; even a guarantee of correctness will not impress them unless the system lets them understand and tinker with their formal proofs.

We present a formalization of convex polyhedra in the proof assistant Coq. The cornerstone of our work is a complete implementation of the simplex method, together with the proof of its correctness and termination. This allows us to define the basic predicates over polyhedra in an effective way (i.e., as programs), and relate them with the corresponding usual logical counterparts. To this end, we make an extensive use of the Boolean reflection methodology. The benefit of this approach is that we can easily derive the proof of several fundamental results on polyhedra, such as Farkas' Lemma, the duality theorem of linear programming, and Minkowski's Theorem., 18 pages, 2 figures, extended version

The number balancing (NBP) problem is the following: given real numbers \(a_1,\ldots ,a_n \in [0,1]\), find two disjoint subsets \(I_1,I_2 \subseteq [n]\) so that the difference \(|\sum _{i \in I_1}a_i - \sum _{i \in I_2}a_i|\) of their sums is minimized. An application of the pigeonhole principle shows that there is always a solution where the difference is at most \(O(\frac{\sqrt{n}}{2^n})\). Finding the minimum, however, is NP-hard. In polynomial time, the differencing algorithm by Karmarkar and Karp from 1982 can produce a solution with difference at most \(n^{-\varTheta (\log n)}\), but no further improvement has been made since then.