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From the zonotope construction to the Minkowski addition of convex polytopes

Komei Fukuda
2004 Journal of symbolic computation  
By replacing line segments with convex V-polytopes, we obtain a natural generalization of the zonotope construction problem: the construction of the Minkowski addition of k polytopes.  ...  A zonotope is the Minkowski addition of line segments in R d . The zonotope construction problem is to list all extreme points of a zonotope given by its line segments.  ...  The main objective of the present paper is to introduce an polynomial algorithm for the Minkowski addition of k convex polytopes in R d .  ... 
doi:10.1016/j.jsc.2003.08.007 fatcat:gxreubwvyfa5tixdykrg5dg2ma

A linear optimization oracle for zonotope computation [article]

Antoine Deza, Lionel Pournin
2019 arXiv   pre-print
A variation of the latter algorithm also allows to decide whether a polytope, given as its vertex set, is a zonotope and when it is not, to compute its greatest zonotopal summand.  ...  More precisely, our algorithms compute the vertices of a zonotope from the set of its generators and inversely, recover the generators of a zonotope from its vertices.  ...  By construction, Z X is the Minkowski sum of its generators. Hence, Z X is the convex hull of all the possible subsums of [−X ] ∪ [G\X ].  ... 
arXiv:1912.02439v1 fatcat:yafh5pjxwbg5regs45snpbolry

Representation of Polytopes as Polynomial Zonotopes [article]

Niklas Kochdumper, Matthias Althoff
2019 arXiv   pre-print
We prove that each bounded polytope can be represented as a polynomial zonotope, which we refer to as the Z-representation of polytopes.  ...  In addition, the Z-representation enables the computation of linear maps, Minkowski addition, and convex hull with a computational complexity that is polynomial in the representation size.  ...  Acknowledgements The authors gratefully acknowledge financial support by the German Research Foundation (DFG) project faveAC under grant AL 1185/5 1.  ... 
arXiv:1910.07271v1 fatcat:nhe3be7qnrhitftkwbrd2sn75e

Zonotope bundles for the efficient computation of reachable sets

Matthias Althoff, Bruce H. Krogh
2011 IEEE Conference on Decision and Control and European Control Conference  
Reachable set computations suffer from the curse of dimensionality, which has been successfully addressed by using zonotopes for linear systems.  ...  We introduce zonotope bundles as the intersection of zonotopes (without explicitly computing the intersection).  ...  Fig. 1 . 1 Possible representations of a polytope. Fig. 2 . 2 Construction of a zonotope by Minkowski addition of line segments.  ... 
doi:10.1109/cdc.2011.6160872 dblp:conf/cdc/AlthoffK11 fatcat:esxwhfjgqbfp3m5gsdtj4zqwi4

Ellipsotopes: Combining Ellipsoids and Zonotopes for Reachability Analysis and Fault Detection [article]

Shreyas Kousik, Adam Dai, Grace Gao
2022 arXiv   pre-print
Zonotopes, a type of symmetric, convex polytope, are similarly frequently used due to efficient numerical implementation of affine maps and exact Minkowski sums.  ...  Ellipsotopes combine the advantages of ellipsoids and zonotopes while ensuring convex collision checking. The utility of this representation is demonstrated on several examples.  ...  A zonotope is a centrally-symmetric convex polytope constructed as a Minkowski sum of line segments.  ... 
arXiv:2108.01750v4 fatcat:hedmsjlxbrgzhgp6vzextos2du

Page 318 of Mathematical Reviews Vol. , Issue 96a [page]

1996 Mathematical Reviews  
In Section 2 the author treats a variant of the notion of indecomposability with respect to Minkowski addition; this connects with CB, Section 3.2.  ...  A zonotope is a polytope which can be expressed as a Minkowski sum of a finite number of closed segments or, equivalently, a poly- tope all of whose faces are centrally symmetric.  ... 

Guaranteed State Estimation for Nonlinear Discrete-Time Systems via Indirectly Implemented Polytopic Set Computation

Jian Wan, Sanjay Sharma, Robert Sutton
2018 IEEE Transactions on Automatic Control  
by the intersection of zonotopes.  ...  The new approach keeps the polytopic set resulting from the intersection intact and computes the evolution of this intact polytopic set for the next time step through representing the polytopic set exactly  ...  A relatively efficient algorithm was proposed in [21] to address the zonotope construction problem, where the addition of line segments was replaced by the addition of convex polytopes.  ... 
doi:10.1109/tac.2018.2816262 fatcat:vabztftv6zbm5lv653s2nef52e

A Brief Survey on Lattice Zonotopes [article]

Benjamin Braun, Andrés R. Vindas-Meléndez
2018 arXiv   pre-print
Zonotopes are a rich and fascinating family of polytopes, with connections to many areas of mathematics.  ...  In this article we provide a brief survey of classical and recent results related to lattice zonotopes. Our emphasis is on connections to combinatorics, both in the sense of enumeration (e.g.  ...  Thanks to Federico Ardila, Matt Beck, and Sam Hopkins for helpful comments.  ... 
arXiv:1808.04933v2 fatcat:i4mhqk7u2rbppnsjssgpca6flu

Primitive Zonotopes [article]

Antoine Deza, George Manoussakis, Shmuel Onn
2017 arXiv   pre-print
We introduce and study a family of polytopes which can be seen as a generalization of the permutahedron of type B_d.  ...  We highlight connections with the largest possible diameter of the convex hull of a set of points in dimension d whose coordinates are integers between 0 and k, and with the computational complexity of  ...  out graphical zonotopes and that Z 1 (d, 2) is the permutahedron of type B d .  ... 
arXiv:1512.08018v4 fatcat:ldn6xws2jjevnoy6i5vfi4sjmq

Primitive Zonotopes

Antoine Deza, George Manoussakis, Shmuel Onn
2017 Discrete & Computational Geometry  
We introduce and study a family of polytopes which can be seen as a generalization of the permutahedron of type B d .  ...  We highlight connections with the largest possible diameter of the convex hull of a set of points in dimension d whose coordinates are integers between 0 and k, and with the computational complexity of  ...  out graphical zonotopes and that Z 1 (d, 2) is the permutahedron of type B d .  ... 
doi:10.1007/s00454-017-9873-z fatcat:aoipv7quabdh3kpqm2gwrock5y

On $${\pi}$$ π -Surfaces of Four-Dimensional Parallelohedra

Alexey Garber
2017 Annals of Combinatorics  
Namely we show that for every four-dimensional parallelohedron P the group of rational first homologies of its \pi-surface is generated by half-belt cycles.  ...  We show that every four-dimensional parallelohedron P satisfies a recently found condition of Garber, Gavrilyuk & Magazinov sufficient for the Voronoi conjecture being true for P.  ...  A convex polytope Z ⊂ R d is called zonotope if it can be represented as a Minkowski sum of finite number of segments. Equivalently, any zonotope Z is a projection of some cube of dimension n ≥ d.  ... 
doi:10.1007/s00026-017-0366-9 fatcat:gy4yrv6urfg5xfqvyowtiyc6am

Aggregation and Disaggregation of Energetic Flexibility from Distributed Energy Resources [article]

Fabian L. Müller, Jácint Szabó, Olle Sundström, John Lygeros
2017 arXiv   pre-print
The description proposed allows aggregators to efficiently pool the flexibility of large numbers of systems and to make control and market decisions on the aggregate level.  ...  To fully leverage the flexibility available from distributed small-scale resources, their flexibility must be quantified and aggregated.  ...  ACKNOWLEDGMENT The authors gratefully acknowledge the fruitful discussions with Stefan Wörner, Ulrich Schimpel, and Marco Laumanns.  ... 
arXiv:1705.02815v1 fatcat:2ejuczpvrfep3gsljl7n44bbry

The Cayley Trick, lifting subdivisions and the Bohne-Dress theorem on zonotopal tilings

Birkett Huber, Jörg Rambau, Francisco Santos
2000 Journal of the European Mathematical Society (Print)  
As an application, we show that the Cayley Trick combined with results of Santos on subdivisions of Lawrence polytopes provides a new independent proof of the Bohne-Dress theorem on zonotopal tilings.  ...  In 1994, Sturmfels gave a polyhedral version of the Cayley Trick of elimination theory: he established an order-preserving bijection between the posets of coherent mixed subdivisions of a Minkowski sum  ...  r there are the following isomorphisms of posets: On the level of convex hulls the above representation for the Minkowski sum polytope is nothing else but the ordinary intersection of the Cayley  ... 
doi:10.1007/s100970050003 fatcat:qpno3jmpozfzlltopa2nianfeq

Sparse Polynomial Zonotopes: A Novel Set Representation for Reachability Analysis [article]

Niklas Kochdumper, Matthias Althoff
2019 arXiv   pre-print
In addition, we can significantly reduce the computation time compared to zonotopes.  ...  Sparse polynomial zonotopes can represent non-convex sets and are generalizations of zonotopes and Taylor models.  ...  However, the disadvan- tage of ellipsoids is that they are not closed under intersection and Minkowski addition; the disadvantage of polytopes is that Minkowski sum is computationally expensive [37] .  ... 
arXiv:1901.01780v1 fatcat:bwnbolpxzreeldreuctwg5kuba

Structure results for multiple tilings in 3D [article]

Nick Gravin, Mihail Kolountzakis, Sinai Robins, Dmitry Shiryaev
2012 arXiv   pre-print
This exceptional class consists of two-flat zonotopes P, defined by the Minkowski sum of n+m line segments that lie in the union of two different two-dimensional subspaces H_1 and H_2.  ...  Equivalently, a two-flat zonotope P may be thought of as the Minkowski sum of two 2-dimensional symmetric polygons one of which may degenerate into a single line segment.  ...  To describe this class, we first recall the definition of a zonotope, which is the Minkowski sum of a finite number of line segments.  ... 
arXiv:1208.1439v1 fatcat:xypkaxmf2bantay6uh2s3zqy5e
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