Search

We introduce a theory of volume polynomials and corresponding duality algebras of multi-fans. Any complete simplicial multi-fan $\Delta$ determines a volume polynomial $V_\Delta$ whose values are the volumes of multi-polytopes based on $\Delta$. This homogeneous polynomial is further used to construct a Poincare duality algebra $\mathcal{A}^*(\Delta)$. We study the structure and properties of $V_\Delta$ and $\mathcal{A}^*(\Delta)$ and give applications and connections to other subjects, such as Macaulay duality, Novik--Swartz theory of face rings of simplicial manifolds, generalizations of Minkowski's theorem on convex polytopes, cohomology of torus manifolds, computations of volumes, and linear relations on the powers of linear forms. In particular, we prove that the analogue of the $g$-theorem does not hold for multi-polytopes., Comment: 45 pages, 3 figures

A new method is presented for computing the Minkowski sum outer-face of two polygons of any shape. Stemming from the contributing vertex concept, the concept of generalized contributing vertex is proposed. Based on the new concept, an efficient algorithm is developed, which starts from the construction of the superset of the Minkowski sum edges. The superset is composed of three types of edges: translated-corner edges, translated edges and corner edges. Then the Minkowski sum outer-face is extracted from the arrangement of the superset edges. The algorithm is implemented using C++ and the Computational Geometry Algorithms Library (CGAL). The experiments including very complicated polygons are conducted, suggesting that the proposed algorithm is more efficient than other existing algorithms in most cases.

We prove that for smooth projective toric varieties, the Okounkov body of a $T$-invariant pseudo-effective divisor with respect to a $T$-invariant flag decomposes as a finite Minkowski sum of indecomposable polytopes. We prove that these indecomposable polytopes form a Minkowski base and that they correspond to the rays in the secondary fan. Moreover, we present an algorithm to find the Minkowski base., Version 2: one more section on how to perform a Minkowski decomposition for toric polytopes Version 3: minor changes

We derive tight expressions for the maximum number of $k$-faces, $0\le{}k\le{}d-1$, of the Minkowski sum, $P_1+...+P_r$, of $r$ convex $d$-polytopes $P_1,...,P_r$ in $\mathbb{R}^d$, where $d\ge{}2$ and $r, Comment: 43 pages; minor changes (mostly typos)

The paper discusses the properties of four-dimensional Minkowski space by using Clifford algebra, then gives the concept of hyperbolic harmonic function by constructing a system (P4) in the four-dimensional Minkowski space, and obtains several properties and a sufficient and necessary condition for the solvability of the system (P4) .

We consider the problem of lower bounding a generalized Minkowski measure of subsets of a convex body with a log-concave probability measure, conditioned on the set size. A bound is given in terms of diameter and set size, which is sharp for all set sizes, dimensions, and norms. In the case of uniform density a stronger theorem is shown which is also sharp.

We give generalizations of Jung's theorem for the case when there are two Minkowski metrics in ℝ n one of which is used to define the diameter of a set M ∈ ℝ n , and the other to determine the radius of the Minkowski ball containing M.

Recently, Dekster introduced a new angle measure for Minkowski spaces according to which the total angular measure τ around a point in a two-dimensional Minkowski space need not be 2π. In this paper, we shall show that while τ need not be 2π, τ always lies between \(\sqrt{2\pi}\) and 8.

We generalize the notion of successive minima, Minkowski?s second theorem and Siegel?s lemma to a free module over a simple algebra whose center is a global field

The classical Minkowski sum of convex sets is defined by the sum of the corresponding support functions. The Lp-extension of such a definition makes use of the sum of the pth power of the support functions. An Lp-zonotope Zp is the p-sum of finitely many segments and is isometric to the unit ball of a subspace of lq, where 1/p + 1/q = 1. In this paper, a sharp upper estimate is given of the volume of Zp in terms of the volume of Z1, as well as a sharp lower estimate of the volume of the polar of Zp in terms of the same quantity. In particular, for p = 1, the latter result provides a new approach to Reisner's inequality for the Mahler conjecture in the class of zonoids.

In the Euclidean plane ℝ2, we define the Minkowski difference 𝒦–𝓛 of two arbitrary convex bodies 𝒦, 𝓛 as a rectifiable closed curve ℋh⊂ ℝ2that is determined by the differenceh=h𝒦–h𝓛of their support functions. This curve ℋhis called the hedgehog with support functionh. More generally, the object of hedgehog theory is to study the Brunn–Minkowski theory in the vector space of Minkowski differences of arbitrary convex bodies of Euclidean space ℝn+1, defined as (possibly singular and self-intersecting) hypersurfaces of ℝn+1. Hedgehog theory is useful for: (i) studying convex bodies by splitting them into a sum in order to reveal their structure; (ii) converting analytical problems into geometrical ones by considering certain real functions as support functions. The purpose of this paper is to give a detailed study of plane hedgehogs, which constitute the basis of the theory. In particular: (i) we study their lengthmeasures and solve the extension of the Christoffel–Minkowski problemto plane hedgehogs; (ii) we characterize support functions of plane convex bodies among support functions of plane hedgehogs and support functions of plane hedgehogs among continuous functions; (iii) we study the mixed area of hedgehogs in ℝ2and give an extension of the classical Minkowski inequality (and thus of the isoperimetric inequality) to hedgehogs.

We prove existence and uniqueness of entire spacelike hypersurfaces in the Minkowski space with prescribed negative scalar curvature, and with given values at infinity which stay at a bounded distance of a lightcone., 21 pages; revised version; to appear in Calc. of Var. and PDE's

In the paper, the analog of Vahlen's theorem for Minkowski bases of three-dimensional lattices is sharpened.

In this work we show that on the space of solutions of a certain class of fourth-order ODEs, u'''' = Λ(s, u, u', u'', u'''), a four-dimensional conformal metric, gab, can be constructed such that the four-dimensional eikonal equation, gabu,au,b = 0, holds. Furthermore, we remark that this structure is invariant under contact transformations. Our general results are applied to the Minkowski spacetime.

The dependence of the complete upper angle in the sense of A. D. Aleksandrov about a point on the Minkowski plane on the form of the “unit circle” (the centrally symmetric convex curve Φ determining the Minkowski metric ρΦ) is studied.The complete upper angle is computed in three cases: if Φ is a square, a “cut circle,” or a “rounded rhombus.” Bibliography: 6 titles.

Minkowski geometric algebra is concerned with sets in the complex plane that are generated by algebraic combinations of complex values varying independently over given sets in ℂ. This algebra provides an extension of real interval arithmetic to sets of complex numbers, and has applications in computer graphics and image analysis, geometrical optics, and dynamical stability analysis. Algorithms to compute the boundaries of Minkowski sets usually invoke redundant segmentations of the operand-set boundaries, guided by a “matching” criterion. This generates a superset of the true Minkowski set boundary, which must be extracted by the laborious process of identifying and culling interior edges, and properly organizing the remaining edges. We propose a new approach, whereby the matching condition is regarded as an implicit curve in the space ℝn whose coordinates are boundary parameters for the n given sets. Analysis of the topological configuration of this curve facilitates the identification of sets of segments on the operand boundaries that generate boundary segments of the Minkowski set, and rejection of certain sets that satisfy the matching criterion but yield only interior edges. Geometrical relations between the operand set boundaries and the implicit curve in ℝn are derived, and the use of the method in the context of Minkowski sums, products, planar swept volumes, and Horner terms is described.

It is shown that every compact convex set K which is centrally symmetric and has a non-empty interior admits a lattice packing of Euclidean 3-space with density greater than or equal to 0.53835.... This is an improvement of the result in [8], which achieved a bound of 0.46421.... Minkowski combinations and the Brunn-Minkowski inequality are used in conjunction with the construction in [8] to achieve a better result.

We prove a Brunn–Minkowski-type inequality for the eigenvalue Λ of the Monge–Ampere operator: Λ - 1 / 2 n is concave in the class of C + 2 domains in R n endowed with Minkowski addition. The equality case is explicitly described too. The main device of the proof is a notion of addition for convex functions, called infimal convolution, which corresponds to the Minkowski addition of the graphs of the involved functions.

We present a view of log-concave measures, which enables one to build an isomorphic theory for high dimensional log-concave measures, analogous to the corresponding theory for convex bodies. Concepts such as duality and the Minkowski sum are described for log-concave functions. In this context, we interpret the Brunn–Minkowski and the Blaschke–Santalo inequalities and prove the two corresponding reverse inequalities. We also prove an analog of Milman’s quotient of subspace theorem, and present a functional version of the Urysohn inequality.

The properties of the symmetry and ordering of Minkowski plane are discussed by using hyperbolic imaginary unit and elliptic imaginary unit of Clifford algebra, and the representations of Cauchy-Riemann equations are given in Minkowski plane.

Two new approaches are presented to establish the existence of polytopal solutions to the discrete-data Lp Minkowski problem for all p > 1.

Generalizing a result of A. Heppes [Hep59] we obtain the following characterization theorem: a convex body K in a Minkowski plane (i.e., in a real, two-dimensional Banach space) is of constant Minkowskian width if and only if every chord of it splits K into two compact sets such that one of them has diameter equal to the length of this chord.

The traditional solution to the Minkowski problem for polytopes involves two steps. First, the existence of a polytope satisfying given boundary data is demonstrated. In the second step, the uniqueness of that polytope (up to translation) is then shown to follow from the equality conditions of Minkowski's inequality, a generalized isoperimetric inequality for mixed volumes that is typically proved in a separate context. In this article we adapt the classical argument to prove both the existence theorem of Minkowski and his mixed volume inequality simultaneously, thereby providing a new proof of Minkowski's inequality that demonstrates the equiprimordial relationship between these two fundamental theorems of convex geometry.

As a generalization of Minkowski sums, products, powers, and roots of complex sets, we consider the Minkowski value of a given polynomial P over a complex set X. Given any polynomial P(z) with prescribed coefficients in the complex variable z, the Minkowski value P(X) is defined to be the set of all complex values generated by evaluating P, through a specific algorithm, in such a manner that each instance of z in this algorithm varies independently over X. The specification of a particular algorithm is necessary, since Minkowski sums and products do not obey the distributive law, and hence different algorithms yield different Minkowski value sets P(X). When P is of degree n and X is a circular disk in the complex plane we study, as canonical cases, the Minkowski monomial value P m (X), for which the monomial terms are evaluated separately (incurring $$ \frac{1}{2} $$ n(n+1) independent values of z) and summed; the Minkowski factor value P f (X), where P is represented as the product (z−r 1)⋅⋅⋅(z−r n ) of n linear factors – each incurring an independent choice z∈X – and r 1,...,r n are the roots of P(z); and the Minkowski Horner value P h (X), where the evaluation is performed by “nested multiplication” and incurs n independent values z∈X. A new algorithm for the evaluation of P h (X), when 0∉X, is presented.

The paper gives a new formulation of the Minkowski sum of polygons. In the conventional Minkowski sum, the inverse operation is not well defined unless the polygons are restricted to be convex. In the proposed formulation, on the other hand, the set of polygons is extended to the set of "hyperpolygons", in which the Minkowski sum forms a commutative group. Consequently, every polygon has its unique inverse, and the sum and the inverse operations can be taken freely. A relation between hyperpolygons and physical reality is also discussed.

We prove that the affine deformation of a Minkowski norm is a Minkowski norm. Some important classical norms are derived by using the affine deformation recipe. Applications are done for some special Finsler manifolds.

A volume-normalized formulation of the Lp-Minkowski problem is presented. This formulation has the advantage that a solution is possible for all p > 1, including the degenerate case where the index p is equal to the dimension of the ambient space. A new approach to the Lp-Minkowski problem is presented, which solves the volume-normalized formulation for even data and all p ≥ 1.

In this paper, we get the measure of strips and lines in the Minkowski plane that meet a fixed compact convex body in . From this we also investigate the probability that strips and lines meet a fixed compact convex body in .

We extend the notion of a double normal of a convex body from smooth, strictly convex Minkowski planes to arbitrary two-dimensional real, normed, linear spaces in two different ways. Then, for both of these ways, we obtain the following characterization theorem: a convex body K in a Minkowski plane is of constant Minkowskian width iff every chord I of K splits K into two compact convex sets K1 and K2 such that I is a Minkowskian double normal of K1 or K2. Furthermore, the Euclidean version of this theorem yields a new characterization of d-dimensional Euclidean ball where d ≥ 3.

Extending our previous work [5] on this subject we show global existence of smooth solutions to the Cauchy problem for wave maps from the (1 + 2)-dimensional Minkowski space to an arbitrary smooth, compact Riemannian manifold without boundary, for arbitrary smooth, radially symmetric data.

Functional digraphs are sometimes fractal sets. As a special kind of fractal sets, the dimension properties of the functional digraph are studied in this paper. Firstly, the proof of a Minkowski dimension theorem is discussed and a new proof is given. Secondly, according to this dimension theorem, the Minkowski dimensions of the digraphs of the sum, deviation, product and quotient of two functions are discussed. And the relations between these Minkowski dimensions and the Minkowski dimensions of the digraphs of the two functions are established. In the conclusion, the maximum Minkowski dimension of the two functional digraphs plays a decisive part in the Minkowski dimensions of the digraphs of the sum, deviation, product and quotient of two functions.

Clifford algebraic geometry corresponds to Minkowski space. Using the discrete structure of Minkowski space, we can abstract a class n-dimensional hyperbolic Hilbert phase space. To discussing the causality of physical event in Minkowski space, we can obtain the geometrical interpretation of uncertainty relation.

We give a differential geometric interpretation for the classification of biharmonic curves in semi-Euclidean3-space due to Chen and Ishikawa (1991).

We present a formalism for dimensional reduction based on the local properties of invariant cross-sections (“fields”) and differential operators. This formalism does not need an ansatz for the invariant fields and is convenient when the reducing group is non-compact. In the approach presented here, splittings of some exact sequences of vector bundles play a key role. In the case of invariant fields and differential operators, the invariance property leads to an explicit splitting of the corresponding sequences, i.e. to the reduced field/operator. There are also situations when the splittings do not come from invariance with respect to a group action but from some other conditions, which leads to a “non-canonical” reduction. In a special case, studied in detail in the second part of this article, this method provides an algorithm for construction of conformally invariant fields and differential operators in Minkowski space.

A model for strings in Minkowski space based on Konopelchenko's generalized representation of surfaces is presented. It is shown how a dynamical equation for the string coordinates can be obtained from the Nambu–Goto action. A class of exponential solutions to the combined Konopelchenko system and dynamical equations are obtained. These can be used to write general solutions of the wave equation composed of right and left moving solutions by using the superposition principle.

We consider the number of integral solutions to the inequality | F ( x ) | ≤ m |F(\mathbf {x}) |\le m , where F ( X ) ∈ Z [ X ] F(\mathbf {X} )\in \mathbb {Z} [\mathbf {X} ] is a decomposable form of degree n + 1 n+1 in n n variables. We show that the number of such solutions is finite for all m m only if the discriminant of F F is not zero. We get estimates for the number of such solutions that display appropriate behavior in terms of the discriminant. These estimates sharpen recent results of the author for the general case of arbitrary degree.

In this paper, the properties ofn-dimensional Minkowski space discussed by using Clifford algebra. The hyperbolic Euler formula is given inn-dimensional Minkowski space.

An invariant form of the Rytov-Vladimirsky law written in an arbitrary curvilinear frame of reference was proposed. The law was generalized to the four-dimensional Minkowski space. The applied problem of temporal evolution of photon polarization was considered. A geometrical interpretation of the generalized Rytov-Vladimirsky law was discussed.

An exact parameterization for the boundary of the Minkowski product of N circular disks in the complex plane is derived. When N> 2, this boundary curve may be regarded as a generalization of the Cartesian oval that bounds the Minkowski product of two disks. The derivation is based on choosing a system of coordinated polar representations for the N operands, identifying sets of corresponding points with matched logarithmic Gauss map that may contribute to the Minkowski product boundary. By means of inversion in the operand circles, a geometrical characterization for their corresponding points is derived, in terms of intersections with the circles of a special coaxal system. The resulting parameterization is expressed as a product of N terms, each involving the radius of one disk, a single square root, and the sine and cosine of a common angular variable ϕ over a prescribed domain. As a special case, the N-th Minkowski power of a single disk is bounded by a higher trochoid. In certain applications, the availability of exact Minkowski products is a useful alternative to the naive bounding approximations that are customarily employed in "complex circular arithmetic." 1. Preamble

In this paper, we introduce pseudo-perpendicular vectors in Minkowski plane and give geometric interpretations of the oriented pseudo-angles in terms of the hyperbolic arcs of finite hyperbolic lengths.

In this paper, we study the so-called mean Minkowski measures, proposed and studied by Toth in a series of papers, for convex bodies of constant width. We show that, with respect to the mean Minkowski measure, the completions of regular simplices are, as well as for many other measures, the most asymmetric ones among all convex bodies of constant width.