87 research outputs found
On the sum of the Voronoi polytope of a lattice with a zonotope
A parallelotope is a polytope that admits a facet-to-facet tiling of
space by translation copies of along a lattice. The Voronoi cell
of a lattice is an example of a parallelotope. A parallelotope can be
uniquely decomposed as the Minkowski sum of a zone closed parallelotope and
a zonotope , where is the set of vectors used to generate the
zonotope. In this paper we consider the related question: When is the Minkowski
sum of a general parallelotope and a zonotope a parallelotope? We give
two necessary conditions and show that the vectors have to be free. Given a
set of free vectors, we give several methods for checking if is
a parallelotope. Using this we classify such zonotopes for some highly
symmetric lattices.
In the case of the root lattice , it is possible to give a more
geometric description of the admissible sets of vectors . We found that the
set of admissible vectors, called free vectors, is described by the well-known
configuration of lines in a cubic. Based on a detailed study of the
geometry of , we give a simple characterization of the
configurations of vectors such that is a
parallelotope. The enumeration yields maximal families of vectors, which
are presented by their description as regular matroids.Comment: 30 pages, 4 figures, 4 table
Congruence and Metrical Invariants of Zonotopes
Zonotopes are studied from the point of view of central symmetry and how
volumes of facets and the angles between them determine a zonotope uniquely.
New proofs are given for theorems of Shephard and McMullen characterizing a
zonotope by the central symmetry of faces of a fixed dimension. When a zonotope
is regarded as the Minkowski sum of line segments determined by the columns of
a defining matrix, the product of the transpose of that matrix and the matrix
acts as a shape matrix containing information about the edges of the zonotope
and the angles between them. Congruence between zonotopes is determined by
equality of shape matrices. This condition is used, together with volume
computations for zonotopes and their facets, to obtain results about rigidity
and about the uniqueness of a zonotope given arbitrary normal-vector and
facet-volume data. These provide direct proofs in the case of zonotopes of more
general theorems of Alexandrov on the rigidity of convex polytopes, and
Minkowski on the uniqueness of convex polytopes given certain normal-vector and
facet-volume data. For a zonotope, this information is encoded in the
next-to-highest exterior power of the defining matrix.Comment: 23 pages (typeface increased to 12pts). Errors corrected include
proofs of 1.5, 3.5, and 3.8. Comments welcom
Bounded Error Identification of Systems With Time-Varying Parameters
This note presents a new approach to guaranteed system identification for time-varying parameterized discrete-time systems. A bounded description of noise in the measurement is considered. The main result is an algorithm to compute a set that contains the parameters consistent with the measured output and the given bound of the noise. This set is represented by a zonotope, that is, an affine map of a unitary hypercube. A recursive procedure minimizes the size of the zonotope with each noise corrupted measurement. The zonotopes take into account the time-varying nature of the parameters in a nonconservative way. An example has been provided to clarify the algorithm
A set-membership state estimation algorithm based on DC programming
This paper presents a new approach to guaranteed state estimation for nonlinear discrete-time systems with a bounded description of noise and parameters. The sets of states that are consistent with the evolution of the system, the measured outputs and bounded noise and parameters are represented by zonotopes. DC programming and intersection operations are used to obtain a tight bound. An example is given to illustrate the proposed algorithm.Ministerio de Ciencia y TecnologĂa DPI2006-15476-C02-01Ministerio de Ciencia y TecnologĂa DPI2007-66718-C04-01
Determinants and the volumes of parallelotopes and zonotopes
AbstractThe Minkowski sum of edges corresponding to the column vectors of a matrix A with real entries is the same as the image of a unit cube under the linear transformation defined by A with respect to the standard bases. The geometric object obtained in this way is a zonotope, Z(A). If the columns of the matrix are linearly independent, the object is a parallelotope, P(A). In the first section, we derive formulas for the volume of P(A) in various ways as ATA, as the square root of the sum of the squares of the maximal minors of A, and as the product of the lengths of the edges of P(A) times the square root of the determinant of the matrix of cosines of angles between pairs of edges. In the second section, we use the volume formulas to derive real-case versions of several well-known determinantal inequalities—those of Hadamard, Fischer, Koteljanskii, Fan, and Szasz—involving principal minors of a positive-definite Hermitian matrix. In the last section, we consider zonotopes, obtain a new proof of the decomposition of a zonotope into its generating parallelotopes, and obtain a volume formula for Z(A)
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