Showing papers on "Minkowski addition published in 2016"

Geometric measures in the dual Brunn–Minkowski theory and their associated Minkowski problems

Abstract: A longstanding question in the dual Brunn–Minkowski theory is “What are the dual analogues of Federer’s curvature measures for convex bodies?” The answer to this is provided. This leads naturally to dual versions of Minkowski-type problems: What are necessary and sufficient conditions for a Borel measure to be a dual curvature measure of a convex body? Sufficient conditions, involving measure concentration, are established for the existence of solutions to these problems.

Mirror symmetric solutions to the centro-affine Minkowski problem

Abstract: The centro-affine Minkowski problem, a critical case of the $$L_p$$ -Minkowski problem in the $$n+1$$ dimensional Euclidean space is considered. By applying methods of calculus of variations and blow-up analyses, two sufficient conditions for the existence of solutions to the centro-affine Minkowski problem are established.

Maxwell and Rankine reciprocal diagrams via Minkowski sums for two-dimensional and three-dimensional trusses under load:

Abstract: A method is presented for computing and displaying the Maxwell (two-dimensional) or Rankine (three-dimensional) diagram reciprocal to a truss under load. Reciprocals are constructed via the two-dim...

A Comprehensive Method for Reachability Analysis of Uncertain Nonlinear Hybrid Systems

Abstract: Reachability analysis of nonlinear uncertain hybrid systems, i.e., continuous-discrete dynamical systems whose continuous dynamics, guard sets and reset functions are defined by nonlinear functions, can be decomposed in three algorithmic steps: computing the reachable set when the system is in a given operation mode, computing the discrete transitions, i.e., detecting and localizing when (and where) the continuous flowpipe intersects the guard sets, and aggregating the multiple trajectories that result from an uncertain transition once the whole flow-pipe has transitioned so that the algorithm can resume. This paper proposes a comprehensive method that provides a nicely integrated solution to the hybrid reachability problem. At the core of the method is the concept of MSPB, i.e., geometrical object obtained as the Minkowski sum of a parallelotope and an axes aligned box. MSPB are a way to control the over-approximation of the Taylor's interval integration method. As they happen to be a specific type of zonotope, they articulate perfectly with the zonotope bounding method that we propose to enclose in an optimal way the set of flowpipe trajectories generated by the transition process. The method is evaluated both theoretically by analyzing its complexity and empirically by applying it to well-chosen hybrid nonlinear examples.

Relative Stanley–Reisner theory and Upper Bound Theorems for Minkowski sums

Abstract: In this paper we settle two long-standing questions regarding the combinatorial complexity of Minkowski sums of polytopes: We give a tight upper bound for the number of faces of a Minkowski sum, including a characterization of the case of equality. We similarly give a (tight) upper bound theorem for mixed facets of Minkowski sums. This has a wide range of applications and generalizes the classical Upper Bound Theorems of McMullen and Stanley. Our main observation is that within (relative) Stanley–Reisner theory, it is possible to encode topological as well as combinatorial/geometric restrictions in an algebraic setup. We illustrate the technology by providing several simplicial isoperimetric and reverse isoperimetric inequalities in addition to our treatment of Minkowski sums.

Extracting flexibility of heterogeneous deferrable loads via polytopic projection approximation

Abstract: This paper develops a novel approach to extract the aggregate flexibility of deferrable loads with heterogeneous parameters via polytopic projection approximation. First, an exact characterization of their aggregate flexibility is derived analytically, which in general contains exponentially many inequality constraints with respect to the number of loads. In order to have a tractable solution, we develop a numerical algorithm that gives a sufficient approximation of the exact aggregate flexibility. Geometrically, the flexibility of each individual load is a polytope and their aggregation is the Minkowski sum of these polytopes. Our method is motivated by an alternative interpretation of the Minkowski sum as a projection operation. The aggregate flexibility can be viewed as the projection of a high-dimensional polytope onto the subspace representing the aggregate power. We formulate a robust optimization problem to optimally approximate the polytopic projection with respect to the homothets of a given polytope. To enable efficient and parallel computation of the aggregate flexibility for a large number of loads, a muti-stage aggregation strategy is proposed. Finally, an energy arbitrage problem is solved to demonstrate the effectiveness of the proposed method.

Tolerance Analysis With Polytopes in HV-Description

Abstract: This article proposes the use of polytopes in $\mathcal{HV}$-description to solve tolerance analysis problems. Polytopes are defined by a finite set of half-spaces representing geometric, contact or functional specifications. However, the list of the vertices of the polytopes are useful for computing other operations as Minkowski sums. Then, this paper proposes a truncation algorithm to obtain the $\mathcal{V}$-description of polytopes in $\mathbb{R}^n$ from its $\mathcal{H}$-description. It is detailed how intersections of polytopes can be calculated by means of the truncation algorithm. Minkowski sums as well can be computed using this algorithm making use of the duality property of polytopes. Therefore, a Minkowski sum can be calculated intersecting some half-spaces in the dual space. Finally, the approach based on $\mathcal{HV}$-polytopes is illustrated by the tolerance analysis of a real industrial case using the open source software PolitoCAT and politopix.

A Geometric Analysis of the AWGN Channel With a $(\sigma , \rho )$ -Power Constraint

Abstract: In this paper, we consider the additive white Gaussian noise (AWGN) channel with a power constraint called the $(\sigma , \rho )$ -power constraint, which is motivated by energy harvesting communication systems. Given a codeword, the constraint imposes a limit of $\sigma + k \rho $ on the total power of any $k\geq 1$ consecutive transmitted symbols. Such a channel has infinite memory and evaluating its exact capacity is a difficult task. Consequently, we establish an $n$ -letter capacity expression and seek bounds for the same. We obtain a lower bound on capacity by considering the volume of $ {\mathcal{ S}}_{n}(\sigma , \rho ) \subseteq \mathbb {R}^{n}$ , which is the set of all length $n$ sequences satisfying the $(\sigma , \rho )$ -power constraints. For a noise power of $ u $ , we obtain an upper bound on capacity by considering the volume of $ {\mathcal{ S}}_{n}(\sigma , \rho ) \oplus B_{n}(\sqrt {n u })$ , which is the Minkowski sum of $ {\mathcal{ S}}_{n}(\sigma , \rho )$ and the $n$ -dimensional Euclidean ball of radius $\sqrt {n u }$ . We analyze this bound using a result from convex geometry known as Steiner’s formula, which gives the volume of this Minkowski sum in terms of the intrinsic volumes of $ {\mathcal{ S}}_{n}(\sigma , \rho )$ . We show that as the dimension $n$ increases, the logarithm of the sequence of intrinsic volumes of $\{ {\mathcal{ S}}_{n}(\sigma , \rho )\}$ converges to a limit function under an appropriate scaling. The upper bound on capacity is then expressed in terms of this limit function. We derive the asymptotic capacity in the low- and high-noise regime for the $(\sigma , \rho )$ -power constrained AWGN channel, with strengthened results for the special case of $\sigma = 0$ , which is the amplitude constrained AWGN channel.

An outer approximation of the Minkowski sum of convex conic sets with application to demand response

Abstract: Flexible loads can provide services such as load-shifting and regulation to power system operators through demand response. A system operator must know the aggregate capabilities of a load population to use it in scheduling and dispatch routines such as optimal power flow and unit commitment. It is not practical for a system operator to model every single load because it would compromise tractability and require potentially unavailable information. A key challenge for load aggregators is to develop low-order models of load aggregations that system operators can use in their operating routines. In this paper, we develop a simple approximation for loads modeled by linear, second-order cone, and semidefinite constraints. It is an outer approximation of the Minkowski sum, the exact computation of which is intractable. We apply the outer approximation to loads with convex quadratic apparent power constraints and uncertainty modeled with second-order cone constraints.

The condition number of join decompositions

Abstract: The join set of a finite collection of smooth embedded submanifolds of a mutual vector space is defined as their Minkowski sum. Join decompositions generalize some ubiquitous decompositions in multilinear algebra, namely tensor rank, Waring, partially symmetric rank and block term decompositions. This paper examines the numerical sensitivity of join decompositions to perturbations; specifically, we consider the condition number for general join decompositions. It is characterized as a distance to a set of ill-posed points in a supplementary product of Grassmannians. We prove that this condition number can be computed efficiently as the smallest singular value of an auxiliary matrix. For some special join sets, we characterized the behavior of sequences in the join set converging to the latter's boundary points. Finally, we specialize our discussion to the tensor rank and Waring decompositions and provide several numerical experiments confirming the key results.

Optimality Conditions in Quasidifferentiable Vector Optimization

Abstract: In the paper, the quasidifferentiable vector optimization problem with the inequality constraints is considered. The Fritz John-type necessary optimality conditions and the Karush---Kuhn---Tucker-type necessary optimality conditions for a weak Pareto solution are derived for such a nonsmooth vector optimization problem. Further, the concept of an F-convex function with respect to a convex compact set is introduced. Then, the sufficient optimality conditions for a (weak) Pareto optimality of a feasible solution are established for the considered nonsmooth multiobjective optimization problem under assumptions that the involved functions are quasidifferentiable F-convex with respect to convex compact sets which are equal to Minkowski sum of their subdifferentials and superdifferentials at this point.

Continuous penetration depth computation for rigid models using dynamic Minkowski sums

Abstract: We present a novel, real-time algorithm for computing the continuous penetration depth (CPD) between two interpenetrating rigid models bounded by triangle meshes. Our algorithm guarantees gradient continuity for the penetration depth (PD) results, unlike conventional penetration depth (PD) algorithms that may have directional discontinuity due to the Euclidean projection operator involved with PD computation. Moreover, unlike prior CPD algorithms, our algorithm is able to handle an orientation change in the underlying model and deal with a topologically-complicated model with holes. Given two intersecting models, we interpolate tangent planes continuously on the boundary of the Minkowski sums between the models and find the closest point on the boundary using Phong projection. Given the high complexity of computing the Minkowski sums for polygonal models in 3D, our algorithm estimates a solution subspace for CPD and dynamically constructs and updates the Minkowski sums only locally in the subspace. We implemented our algorithm on a standard PC platform and tested its performance in terms of speed and continuity using various benchmarks of complicated rigid models, and demonstrated that our algorithm can compute CPD for general polygonal models consisting of tens of thousands of triangles with a hole in a few milli-seconds while guaranteeing the continuity of PD gradient. Moreover, our algorithm can compute more optimal PD values than a state-of-the-art PD algorithm due to the dynamic Minkowski sum computation.

Maximally reducible monodromy of bivariate hypergeometric systems

Abstract: We investigate branching of solutions to holonomic bivariate hypergeomet- ric systems of Horn type. Special attention is paid to the invariant subspace of Puiseux polynomial solutions. We mainly study (1) Horn systems defined by simplicial configura- tions, (2) Horn systems whose Ore-Sato polygon is either a zonotope or a Minkowski sum of a triangle and segments. We prove a necessary and sufficient condition for the mon- odromy representation to be maximally reducible, that is, for the space of holomorphic solutions to split into the direct sum of one-dimensional invariant subspaces.

Successive radii and Orlicz Minkowski sum

Abstract: In this paper, we deal with the successive inner and outer radii with respect to Orlicz Minkowski sum. The upper and lower bounds for the radii of the Orlicz Minkowski sum of two convex bodies are established.

m-Projections involving Minkowski inverse and range symmetric property in Minkowski space

Abstract: In this paper we study the impact of Minkowski metric matrix on a projection in the Minkowski Space M along with their basic algebraic and geometric properties.The relation between the m-projections and the Minkowski inverse of a matrix A in the minkowski space M is derived. In the remaining portion commutativity of Minkowski inverse in Minkowski Space M is analyzed in terms of m-projections as an analogous development and extension of the results on EP matrices.

Interval Superposition Arithmetic

Abstract: This paper presents a novel set-based computing method, called interval superposition arithmetic, for enclosing the image set of multivariate factorable functions on a given domain. In order to construct such enclosures, the proposed arithmetic operates over interval superposition models which are parameterized by a matrix with interval components. Every point in the domain of a factorable function is then associated with a sequence of components of this matrix and the superposition, i.e. Minkowski sum, of these elements encloses the image of the function at this point. Interval superposition arithmetic has a linear runtime complexity with respect to the number of variables. Besides presenting a detailed theoretical analysis of the accuracy and convergence properties of interval superposition arithmetic, the paper illustrates its advantages compared to existing set arithmetics via numerical examples.

Eventual quasi-linearity of the Minkowski length

Abstract: 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 t P of a lattice polytope P behaves polynomially in t ź N . In this paper we prove that for any lattice polytope P , the Minkowski length of t P for t ź 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.

Algebra and Geometry of Sets in Boolean Space

Abstract: In the present paper, geometry of the Boolean space Bn in terms of Hausdorff distances between subsets and subset sums is investigated. The main results are the algebraic and analytical expressions for representing of classical figures in Bn and the functions of distances between them. In particular, equations in sets are considered and their interpretations in combinatory terms are given.

Support Theorem for the Light Ray Transform on Minkowski Spaces

Abstract: We study the Light-Ray transform of integrating vector fields on the Minkowski time-space R^{1+n}, n bigger than equal 2, with the Minkowski metric. We prove a support theorem for vector fields vanishing on an open set of light-like geodesics.

Minkowski sum of polytopes and its normality

Abstract: In this paper, we consider the normality or the integer decomposition property (IDP, for short) for Minkowski sums of integral convex polytopes. We discuss some properties on the toric rings associated with Minkowski sums of integral convex polytopes. We also study Minkowski sums of edge polytopes and give a sufficient condition for Minkowski sums of edge polytopes to have IDP.

Smoothness of Minkowski sum and generic rotations

Abstract: Can the Minkowski sum of two compact convex bodies be made smoother by rotating one of them? We construct two infinitely differentiable strictly convex plane bodies such that after any generic rotation (in the Baire category sense) of one of the summands the Minkowski sum is not five times differentiable. On the other hand, if for one of the bodies the zero set of the Gaussian curvature has countable spherical image, we show that any generic rotation makes their Minkowski sum as smooth as the summands. We also improve and clarify some previous results on smoothness of the Minkowski sum.

Partial Ordering of Range Symmetric Matrices and M-Projectors with Respect to Minkowski Adjoint in Minkowski Space

Abstract: 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].

Non-convex Region Description by Hyperplane Arrangements

Abstract: This chapter presents some prerequisites and basic notions which will be instrumental in the rest of the manuscript.

Existence of Minkowski space

Abstract: Minkowski space serves as a framework for the theoretical constructions that deal with manifestations of relativistic effects in physical phenomena. But neither Minkowski himself nor the subsequent developers of the relativity theory have provided a reasonable rationale for this mathematical construct. In physics, such a rationale should show lower-level statements that determine where the proposed mathematical structure is applicable and yield formal premises for proving its existence. The above failure has apparently been due to the features of the adopted formalism based on the unjustifiably exclusive use of coordinates in the theoretical analysis of physical phenomena, which ignores the necessity of having physical grounds for mathematical concepts. In particular, the use of a coordinate transformation between two inertial reference frames makes the consideration so cumbersome that it appears useless for solving the fundamental problems of physical theory, including the question of whether Minkowski space exists. In contrast, a straightforward calculation proves that the transformation of the time and the position vector of a physical event between two physical spaces establishes an equivalence relation between pairs made of these variables. This means the existence of Minkowski space and shows that the premises for its proof are the same as for the coordinate-free derivation of basic effects of the special relativity theory: the use of Einsteinian time variable and motions of particles able to interact with each other and electromagnetic field over a short spatial range only. The high degeneracy of free motions of point particles, together with the intricacy of the above mentioned calculation, suggests that a further generalization of Minkowski space is beyond belief, so that the modification or even the abandonment of the concept of spacetime seems quite natural.

GPU-based visualization of knee-form contact area for safety inspections

Abstract: The United Nations defines a safety regulation based on the possible collision between the driver’s knee and an automobile’s instrument panel. The “knee-form” apparatus used to evaluate compliance with this regulation can be modeled as a Minkowski sum shape of a vertical equilateral triangle and a horizontal cylinder with a radius of 60 mm and a thickness of 120 mm. The knee form contacting condition is geometrically equivalent to that of an equilateral triangle contacting a Minkowski sum shape of the instrument panel and a horizontal cylinder. Based on this concept, we propose a novel algorithm for extracting the knee-form contacting area on the instrument panel. With the parallel computation capability of a Graphics Processing Unit, our system can detect and output the knee-form contacting area in a practical time period.