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From the zonotope construction to the Minkowski addition of convex polytopes

2004
*
Journal of symbolic computation
*

By replacing line segments with

doi:10.1016/j.jsc.2003.08.007
fatcat:gxreubwvyfa5tixdykrg5dg2ma
*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 . ...##
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A linear optimization oracle for zonotope computation
[article]

2019
*
arXiv
*
pre-print

A variation

arXiv:1912.02439v1
fatcat:yafh5pjxwbg5regs45snpbolry
*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 ]. ...##
###
Representation of Polytopes as Polynomial Zonotopes
[article]

2019
*
arXiv
*
pre-print

We prove that each bounded

arXiv:1910.07271v1
fatcat:nhe3be7qnrhitftkwbrd2sn75e
*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. ...##
###
Zonotope bundles for the efficient computation of reachable sets

2011
*
IEEE Conference on Decision and Control and European Control Conference
*

Reachable set computations suffer

doi:10.1109/cdc.2011.6160872
dblp:conf/cdc/AlthoffK11
fatcat:esxwhfjgqbfp3m5gsdtj4zqwi4
*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. ...##
###
Ellipsotopes: Combining Ellipsoids and Zonotopes for Reachability Analysis and Fault Detection
[article]

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. ...

##
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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

2018
*
IEEE Transactions on Automatic Control
*

by

doi:10.1109/tac.2018.2816262
fatcat:vabztftv6zbm5lv653s2nef52e
*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*. ...##
###
A Brief Survey on Lattice Zonotopes
[article]

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. ...

##
###
Primitive Zonotopes
[article]

2017
*
arXiv
*
pre-print

We introduce and study a family

arXiv:1512.08018v4
fatcat:ldn6xws2jjevnoy6i5vfi4sjmq
*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 . ...##
###
Primitive Zonotopes

2017
*
Discrete & Computational Geometry
*

We introduce and study a family

doi:10.1007/s00454-017-9873-z
fatcat:aoipv7quabdh3kpqm2gwrock5y
*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 . ...##
###
On $${\pi}$$ π -Surfaces of Four-Dimensional Parallelohedra

2017
*
Annals of Combinatorics
*

Namely we show that for every four-dimensional parallelohedron P

doi:10.1007/s00026-017-0366-9
fatcat:gy4yrv6urfg5xfqvyowtiyc6am
*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. ...##
###
Aggregation and Disaggregation of Energetic Flexibility from Distributed Energy Resources
[article]

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. ...

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

2000
*
Journal of the European Mathematical Society (Print)
*

As an application, we show that

doi:10.1007/s100970050003
fatcat:qpno3jmpozfzlltopa2nianfeq
*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 ...##
###
Sparse Polynomial Zonotopes: A Novel Set Representation for Reachability Analysis
[article]

2019
*
arXiv
*
pre-print

In

arXiv:1901.01780v1
fatcat:bwnbolpxzreeldreuctwg5kuba
*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] . ...##
###
Structure results for multiple tilings in 3D
[article]

2012
*
arXiv
*
pre-print

This exceptional class consists

arXiv:1208.1439v1
fatcat:xypkaxmf2bantay6uh2s3zqy5e
*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. ...
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