Polytope Research Articles

A Polytope Related to Empirical Distributions, Plane Trees, Parking Functions, and the Associahedron

The volume of the n -dimensional polytope Π n (x):= {y ∈ R n : y i ≥ 0 and y 1 + · · · + y i ≤ x 1 + · · ·+ x i for all 1 ≤ i ≤ n } for arbitrary x:=(x 1 , . . ., x n ) with x i >0 for all i defines a polynomial in variables x i which admits a number of interpretations, in terms of empirical distributions, plane partitions, and parking functions. We interpret the terms of this polynomial as the volumes of chambers in two different polytopal subdivisions of Π n (x) . The first of these subdivisions generalizes to a class of polytopes called sections of order cones. In the second subdivision the chambers are indexed in a natural way by rooted binary trees with n+1 vertices, and the configuration of these chambers provides a representation of another polytope with many applications, the associahedron .

Triangulations of oriented matroids

Introduction Preliminaries on oriented matroids Triangulations of oriented matroids Duality between triangulations and extensions Subdivisions of Lawrence polytopes Lifting triangulations Bibliography.

All-Different Polytopes

All-Different Polytopes

Unimodular covers of multiples of polytopes

Let P be a d -dimensional lattice polytope. We show that there exists a natural number c_d , only depending on d , such that the multiples cP have a unimodular cover for every natural number c\ge c_d . Actually, an explicit upper bound for c_d is provided, together with an analogous result for unimodular covers of rational cones.

A structured algorithm for minimum l∞-norm solutions and its application to a robot velocity workspace analysis

Herein is proposed a concise algorithmic procedure for deriving a minimum l∞-norm solution of the system of consistent linear equations Ax=b, where A is a m×n matrix, b is given as a m×1 vector and x is a n×1 unknown vector. The proposed algorithm is developed based on the geometrical analysis of the characteristics of the minimum infinity-norm solution. The proposed algorithm is well-structured so that it may be implemented easily through simple linear algebraic manipulation. For the case of n>m, the basic idea of the method is extended to finding internally mapped vertices when a n dimensional polytope is transformed into m dimensional polytope through consistent linear mapping A. The proposed method is applied to the task velocity analysis for robot manipulators with joint velocity constraints.

Output Feedback Control with Input Saturations: LMI Design Approaches

This paper addresses the control of linear systems with input saturations. We seek a controller that guarantees for the closed-loop system: (i) stability for a given poly tope of initial conditions, (ii) a prescribed weak L2 gain attenuation between inputs and outputs of interest. Two approaches are proposed based on: (i) ensuring that the controller never saturates: the obtained controller is linear time invariant (LTI), (ii) ensuring absolute stability against the saturations: the controller is then linear time varying (LTV). Existence conditions for these two control structures can be cast as (convex) optimization problems over linear matrix inequalities (LMIs). Finally, using numerical experiments, we compare both approaches. In this numerical examples, the LTI controller presents some advantages.

Besov spaces andK-functionals inL p, 0<p<1*

We investigate Besov spaces and their connection with trigonometric polynomial approximation inLp[−π,π], algebraic polynomial approximation inLp[−1,1], algebraic polynomial approximation inLp(S), and entire function of exponential type approximation inLp(R), and characterizeK-functionals for certain pairs of function spaces including (Lp[−π,π],Bsa(Lp[−π,π])), (Lp(R),sa(Lp(R))),\((L_p [ - 1,1]\overline B _B^\alpha (L_p [ - 1,1])),\), and\((L_p (S),\overline{\overline B} _B^\alpha (L_p (S))),\), where 0<s≤∞, 0<p<1,S is a simple polytope and 0<α<r.

Smith-Type Inequalities for a Polytope with a Solvable Group of Symmetries

The object of this paper is to obtain a set of inequalities relating the face numbers of different orbit types of a simplicial polytope P with a finite solvable group G of linear symmetries. It is assumed that (1) for each subgroup H of G, the fixed point set PH is a subpolytope of P, and (2) the toric variety X(P) associated to P is nonsingular. The action of G on P induces an action on X(P), and we describe a set of Smith-type inequalities between the Betti numbers of X(P)H, where H ranges through the set of subgroups of G. By relating each X(P)H with X(PH), we then express these inequalities in terms of the face numbers of the different orbit types of P and the rank of fixed point sets of certain compact tori. This rank is determined explicitly when G is abelian. Moreover, assumption (2) is removed for a polytope of dimension 2.

Counting faces of cubical spheres modulo two

Several recent papers have addressed the problem of characterizing the f-vectors of cubical polytopes. This is largely motivated by the complete characterization of the f-vectors of simplicial polytopes given by Stanley (Discrete Geometry and Convexity, Annals of the New York Academy of Sciences, Vol. 440, 1985, pp. 212–223) and Billera and Lee (Bull. Amer. Math. Soc. 2 (1980) 181–185) in 1980. Along these lines Blind and Blind (Discrete Comput. Geom. 11(3) (1994) 351–356) have shown that unlike in the simplicial case, there are parity restrictions on the f-vectors of cubical polytopes. In particular, except for polygons, all even dimensional cubical polytopes must have an even number of vertices. Here this result is extended to a class of zonotopal complexes which includes simply connected odd dimensional manifolds. This paper then shows that the only modular equations which hold for the f-vectors of all d-dimensional cubical polytopes (and hence spheres) are modulo two. Finally, the question of which mod two equations hold for the f-vectors of PL cubical spheres is reduced to a question about the Euler characteristics of multiple point loci from codimension one PL immersions into the d-sphere. Some results about this topological question are known (Eccles, Lecture Notes in Mathematics, Vol. 788, Springer, Berlin, 1980, pp. 23–38; Herbert, Mem. Amer. Math. Soc. 34 (250) (1981); Lannes, Lecture Notes in Mathematics, Vol. 1051, Springer, Berlin, 1984, pp. 263–270) and Herbert's result we translate into the cubical setting, thereby removing the PL requirement. A central definition in this paper is that of the derivative complex, which captures the correspondence between cubical spheres and codimension one immersions.

Simple 0/1-Polytopes

For general polytopes, it has turned out that with respect to many questions it suffices to consider only the simple polytopes, i.e., d -dimensional polytopes where every vertex is contained in only d facets. In this paper, we show that the situation is very different within the class of 0/1-polytopes, since every simple 0/1-polytope is the (cartesian) product of some 0/1-simplices (which proves a conjecture of Ziegler), and thus, the restriction to simple 0/1-polytopes leaves only a very small class of objects with a rather trivial structure.

On the Volume of a Certain Polytope

Let n ≥ 2 be an integer and consider the set Tn of n × n permu tation matrices π for wh ich π ij = 0 for j ≥ i + 2. We study the convex hull Pn of Tn, a polytope of dimension (n 2). We provide evidence for several conjectures involving Pn, including Conjecture 1: Let Vn denote the minimum volume of a simplex with vertices in the affine lattice spanned by Tn. Then the volume of Pn is Vn time s the product of the first n – 1 Catalan numbers. We also give a related result on the Ehrhart polynomial of Pn. Editor's note: After this paper was circulated, Doron Zeilberger [1998] proved Conjecture 1, using the authors' reduction of the original problem to a conjectural comb inatorial identity, and sketched the proofs of two others. The problems and methodology presented here gain even further interest thereby.

A Closer Look at Lattice Points in Rational Simplices

We generalize Ehrhart's idea of counting lattice points in dilated rational polytopes: Given a rational simplex, that is, an $n$-dimensional polytope with $n+1$ rational vertices, we use its description as the intersection of $n+1$ halfspaces, which determine the facets of the simplex. Instead of just a single dilation factor, we allow different dilation factors for each of these facets. We give an elementary proof that the lattice point counts in the interior and closure of such a vector-dilated simplex are quasipolynomials satisfying an Ehrhart-type reciprocity law. This generalizes the classical reciprocity law for rational polytopes. As an example, we derive a lattice point count formula for a rectangular rational triangle, which enables us to compute the number of lattice points inside any rational polygon.

Infinitesimal Rigidity of Convex Polytopes

Aleksandrov [1] proved that a simple convex d -dimensional polytope, d ≥ 3 , is infinitesimally rigid if the volumes of its facets satisfy a certain assumption of stationarity. We extend this result by proving that this assumption can be replaced by a stationarity assumption on the k -dimensional volumes of the polytope's k -dimensional faces, where k ∈{1,. . .,d-1} .

An Euler-type volume identity

In this paper we present an elementary proof of a congruence by subtraction relation. In order to prove congruence by subtraction, we produce a dissection relating equal sub-polytopes. An immediate consequence of this relation is an Euler-type volume identity in ℝ3 which appeared in the Unsolved Problems section of the December 1996 MAA Monthly.This Euler-type volume identity relates the volumes of subsets of a polytope called wedges that correspond to its faces, edges, and vertices. A wedge consists of the inward normal chords of the polytope emanating from a face, vertex, or edge. This identity is stated in the theorem below.Euler Volume Theorem. For any three dimensional convex polytope PThis identity follows immediately from

Perfect and Ideal 0, ±1 Matrices

A 0, ±1 matrix A is said to be perfect (resp. ideal) if the corresponding generalized packing (resp. covering) polytope is integral. Given a 0, ±1 matrix A, we construct a 0, 1 matrix that is perfect if and only if A is perfect. A similar result is obtained for the generalized covering problem. We also extend some known results on perfect 0, 1 matrices to the 0, ±1 case.

The number of nets of the regular convex polytopes in dimension ⩽ 4

Classifying the nets (also called unfoldings or developments or patterns) of the regular convex polytopes under the isometry group of the polytope is equivalent to classifying the spanning trees of the facet-adjacency graph under its automorphism group. This is done for all such polytopes of dimension at most 4.

Combinatorially regular Euler polytopes

Combinatorially regular Euler polytopes

An exterior-point polytope sliding and deformation algorithm for linear programming problems

In this research, we develop an algorithm for linear programming problems based on a new interpretation of Karmarkar's representation for this problem. Accordingly, we examine a suitable polytope for which the origin is an exterior point, and in order to determine an optimal solution, we need to ascertain the minimum extent by which this polytope needs to be slid along a one-dimensional axis so that the origin belongs to it. To accomplish this, we employ strongly separating hyperplanes between the origin and the polytope using a closest point routine. The algorithm is further enhanced by the generation of dual solutions which enable us to deform the polytope so that it is favorably positioned with respect to the origin and the axis of sliding motion. The overall scheme is easy to implement, requires a minimal amount of storage, and produces quick good quality lower bounds for the problem in its infinite convergence process. A switchover to the simplex method or an interior point method is also possible, using the current available solution as an advanced start. Preliminary computational results are provided along with implementation guidelines.

Four dimensional quantum topology changes of space–times

Topology changing processes in the WKB approximation of four dimensional quantum cosmology with a negative cosmological constant are investigated. As Riemannian manifolds, which describe quantum tunnelings of space–time, constant negative curvature solutions of the Einstein equation, i.e., hyperbolic geometries are considered. Using four dimensional polytopes, one can explicitly construct hyperbolic manifolds with topologically nontrivial boundaries which describe topology changes. These instantonlike solutions are constructed out of 8-cells, 16-cells, or 24-cells and have several points at infinity called cusps. The hyperbolic manifolds are noncompact because of the cusps but have finite volumes. Topology change amplitudes in the WKB approximation in terms of the volumes of these manifolds are evaluated and it was found that the more complicated the topology changes, the more likely are suppressed.

Irregularities of distributions with respect to polytopes

In the first part of the paper we show that the L2-discrepancy with respect to squares is of the same order of magnitude as the usual L2- discrepancy for point distributions in the K-dimensional torus. In the second part we adapt this method to obtain a generalization of Roth's [7] lower bound (log N)(k-1)/2 (for the usual discrepancy) to the discrepancy with respect to homothetic simple convex poly topes.