scholarly journals Diameter, Decomposability, and Minkowski Sums of Polytopes

2018 ◽  
Vol 62 (4) ◽  
pp. 741-755
Author(s):  
Antoine Deza ◽  
Lionel Pournin
Keyword(s):  
The Other ◽  
Minkowski Sum ◽  
Minkowski Sums ◽  
Bounded Below

AbstractWe investigate how the Minkowski sum of two polytopes affects their graph and, in particular, their diameter. We show that the diameter of the Minkowski sum is bounded below by the diameter of each summand and above by, roughly, the product between the diameter of one summand and the number of vertices of the other. We also prove that both bounds are sharp. In addition, we obtain a result on polytope decomposability. More precisely, given two polytopes $P$ and $Q$, we show that $P$ can be written as a Minkowski sum with a summand homothetic to $Q$ if and only if $P$ has the same number of vertices as its Minkowski sum with $Q$.

COMPUTING MINKOWSKI SUMS OF PLANE CURVES

1995 ◽  
Vol 05 (04) ◽  
pp. 413-432 ◽  
Author(s):  
ANIL KAUL ◽  
RIDA T. FAROUKI
Keyword(s):  
Algebraic Curve ◽  
Numerical Procedure ◽  
The Other ◽  
Minkowski Sum ◽  
Plane Curves ◽  
Parametric Curves ◽  
Implicit Equation ◽  
Minkowski Sums ◽  
High Degree ◽  
The Given

The Minkowski sum of two plane curves can be regarded as the area generated by sweeping one curve along the other. The boundary of the Minkowski sum consists of translated portions of the given curves and/or portions of a more complicated curve, the “envelope” of translates of the swept curve. We show that the Minkowski-sum boundary is describable as an algebraic curve (or subset thereof) when the given curves are algebraic, and illustrate the computation of its implicit equation. However, such equations are typically of high degree and do not offer a practical basis for tracing the boundary. For the case of polynomial parametric curves, we formulate a simple numerical procedure to address the latter problem, based on constructing the Gauss maps of the given curves and using them to identifying “corresponding” curve segments that are to be summed. This yields a set of discretely-sampled arcs that constitutes a superset of the Minkowski-sum boundary, and can be regarded as a planar graph. To extract the true boundary, we present a method for identifying and “trimming” away extraneous arcs by systematically traversing this graph.

scholarly journals (Symplectic) leaves and (5d Higgs) branches in the Poly(go)nesian Tropical Rain Forest

2020 ◽  
Vol 2020 (11) ◽  
Author(s):  
Marieke van Beest ◽  
Antoine Bourget ◽  
Julius Eckhard ◽  
Sakura Schäfer-Nameki
Keyword(s):  
Rain Forest ◽  
Tropical Rain Forest ◽  
Gauge Theories ◽  
Edge Coloring ◽  
Field Theories ◽  
Minkowski Sum ◽  
Coulomb Branch ◽  
Superconformal Field Theories ◽  
Minkowski Sums ◽  
Symplectic Leaves

Abstract We derive the structure of the Higgs branch of 5d superconformal field theories or gauge theories from their realization as a generalized toric polygon (or dot diagram). This approach is motivated by a dual, tropical curve decomposition of the (p, q) 5-brane-web system. We define an edge coloring, which provides a decomposition of the generalized toric polygon into a refined Minkowski sum of sub-polygons, from which we compute the magnetic quiver. The Coulomb branch of the magnetic quiver is then conjecturally identified with the 5d Higgs branch. Furthermore, from partial resolutions, we identify the symplectic leaves of the Higgs branch and thereby the entire foliation structure. In the case of strictly toric polygons, this approach reduces to the description of deformations of the Calabi-Yau singularities in terms of Minkowski sums.

scholarly journals A Tight Lower Bound for Convexly Independent Subsets of the Minkowski Sums of Planar Point Sets

10.37236/484 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Ondřej Bílka ◽  
Kevin Buchin ◽  
Radoslav Fulek ◽  
Masashi Kiyomi ◽  
Yoshio Okamoto ◽  
...  
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Minkowski Sum ◽  
Independent Subset ◽  
Point Sets ◽  
Planar Point ◽  
Minkowski Sums

Recently, Eisenbrand, Pach, Rothvoß, and Sopher studied the function $M(m, n)$, which is the largest cardinality of a convexly independent subset of the Minkowski sum of some planar point sets $P$ and $Q$ with $|P| = m$ and $|Q| = n$. They proved that $M(m,n)=O(m^{2/3}n^{2/3}+m+n)$, and asked whether a superlinear lower bound exists for $M(n,n)$. In this note, we show that their upper bound is the best possible apart from constant factors.

scholarly journals ZEROS OF THE MÖBIUS FUNCTION OF PERMUTATIONS

Mathematika ◽  
2019 ◽  
Vol 65 (4) ◽  
pp. 1074-1092
Author(s):  
Robert Brignall ◽  
Vít Jelínek ◽  
Jan Kynčl ◽  
David Marchant
Keyword(s):  
Möbius Function ◽  
The Other ◽  
Positive Proportion ◽  
Mobius Function ◽  
Bounded Below ◽  
General Conditions

We show that if a permutation $\unicode[STIX]{x1D70B}$ contains two intervals of length 2, where one interval is an ascent and the other a descent, then the Möbius function $\unicode[STIX]{x1D707}[1,\unicode[STIX]{x1D70B}]$ of the interval $[1,\unicode[STIX]{x1D70B}]$ is zero. As a consequence, we prove that the proportion of permutations of length $n$ with principal Möbius function equal to zero is asymptotically bounded below by $(1-1/e)^{2}\geqslant 0.3995$. This is the first result determining the value of $\unicode[STIX]{x1D707}[1,\unicode[STIX]{x1D70B}]$ for an asymptotically positive proportion of permutations $\unicode[STIX]{x1D70B}$. We further establish other general conditions on a permutation $\unicode[STIX]{x1D70B}$ that ensure $\unicode[STIX]{x1D707}[1,\unicode[STIX]{x1D70B}]=0$, including the occurrence in $\unicode[STIX]{x1D70B}$ of any interval of the form $\unicode[STIX]{x1D6FC}\oplus 1\oplus \unicode[STIX]{x1D6FD}$.

Tolerance Analysis With Polytopes in HV-Description

2017 ◽  
Vol 17 (4) ◽  
Author(s):  
Santiago Arroyave-Tobón ◽  
Denis Teissandier ◽  
Vincent Delos
Keyword(s):  
Open Source ◽  
Open Source Software ◽  
Dual Space ◽  
Tolerance Analysis ◽  
Minkowski Sum ◽  
Duality Property ◽  
Finite Set ◽  
Minkowski Sums

This article proposes the use of polytopes in 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 is useful for computing other operations as Minkowski sums. Then, this paper proposes a truncation algorithm to obtain the V-description of polytopes in ℝn from its 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 HV-polytopes is illustrated by the tolerance analysis of a real industrial case using the open source software politocat and politopix.

Applications of convex geometry to Minkowski sums of m ellipsoids in ℝN: Closed-form parametric equations and volume bounds

2021 ◽  
pp. 2140009
Author(s):  
Gregory S. Chirikjian ◽  
Bernard Shiffman
Keyword(s):  
Closed Form ◽  
Convex Geometry ◽  
Convex Bodies ◽  
Minkowski Inequality ◽  
Dimensional Euclidean Space ◽  
Minkowski Sum ◽  
Principal Curvatures ◽  
Reverse Isoperimetric Inequality ◽  
Lower Volume ◽  
Minkowski Sums

General results on convex bodies are reviewed and used to derive an exact closed-form parametric formula for the Minkowski sum boundary of [Formula: see text] arbitrary ellipsoids in [Formula: see text]-dimensional Euclidean space. Expressions for the principal curvatures of these Minkowski sums are also derived. These results are then used to obtain upper and lower volume bounds for the Minkowski sum of ellipsoids in terms of their defining matrices; the lower bounds are sharper than the Brunn–Minkowski inequality. A reverse isoperimetric inequality for convex bodies is also given.

Minkowski Sums: A Simulation Tool for CAD/CAM

1992 ◽  
Author(s):  
Anil Kaul
Keyword(s):  
Path Planning ◽  
Computer Aided Design ◽  
Algebraic Curves ◽  
Minkowski Sum ◽  
Morphological Operations ◽  
Simulation Tool ◽  
Cutter Path ◽  
Cad Cam ◽  
Minkowski Sums ◽  
Deposition Processes

Abstract In this paper we provide a survey of Minkowski sums, their applications as a simulation tool in Computer Aided Design and some simple algorithms to compute them. The intent of the paper is to exhibit the relevance of morphological operations, like Minkowski sums, in practical aspects of CAD/CAM and to provide enough motivation to stimulate research in finding efficient algorithms for their computation. Except for cases where the two initial objects are convex, there are no known algorithms which guarantee efficient performance in the polyhedral domain. Some initial work has been done for finding algorithms where the boundaries of the objects are composed of algebraic curves, but again very little is known on how to actually compute the Minkowski sum for objects bounded by general curves. Some of the applications that we describe in this paper include Robot path planning, interference detection, NC cutter path planning, rounding and filleting, shape design, etching and deposition processes in semiconductor process simulation and polyhedral interpolation.

A MONOTONIC CONVOLUTION FOR MINKOWSKI SUMS

2007 ◽  
Vol 17 (04) ◽  
pp. 383-396 ◽  
Author(s):  
VICTOR MILENKOVIC ◽  
ELISHA SACKS
Keyword(s):  
Crossing Number ◽  
Minkowski Sum ◽  
Running Time ◽  
Crossing Numbers ◽  
Bounded Number ◽  
Input Size ◽  
Inflection Points ◽  
Minkowski Sums ◽  
Planar Regions ◽  
Number Of Segments

We present a monotonic convolution for planar regions A and B bounded by line and circular arc segments. The Minkowski sum equals the union of the cells with positive crossing numbers in the arrangement of the convolution, as is the case for the kinetic convolution. The monotonic crossing number is bounded by the kinetic crossing number, and also by the maximum number of intersecting pairs of monotone boundary chains, which is typically much smaller. We give a Minkowski sum algorithm based on the monotonic convolution. The running time is O (s + nα(n) log (n) + m2), versus O (s + n2) for the kinetic algorithm, with s the input size and with n and m the number of segments in the kinetic and monotonic convolutions. For inputs with a bounded number of turning points and inflection points, the running time is O (sα(s) log s), versus Ω(s2) for the kinetic algorithm. The monotonic convolution is 37% smaller than the kinetic convolution and its arrangement is 62% smaller based on 21 test pairs.

scholarly journals On a Special Class of Hyper-Permutahedra

10.37236/6652 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Geir Agnarsson
Keyword(s):  
Special Class ◽  
Lattice Structure ◽  
Minkowski Sum ◽  
Face Lattice ◽  
Closed Formula ◽  
Minkowski Sums ◽  
Interesting Class ◽  
N Form

Minkowski sums of simplices in ${\mathbb{R}}^n$ form an interesting class of polytopes that seem to emerge in various situations. In this paper we discuss the Minkowski sum of the  simplices $\Delta_{k-1}$ in ${\mathbb{R}}^n$ where $k$ and $n$ are fixed, their flags and some of their face lattice structure. In particular, we derive a closed formula for their exponential generating flag function. These polytopes are simple, include both the simplex $\Delta_{n-1}$ and the permutahedron $\Pi_{n-1}$, and form a Minkowski basis for more general permutahedra. 

Tolerance Analysis With Polytopes in HV-Description

2016 ◽  
Author(s):  
Santiago Arroyave-Tobón ◽  
Denis Teissandier ◽  
Vincent Delos
Keyword(s):  
Open Source ◽  
Open Source Software ◽  
Dual Space ◽  
Tolerance Analysis ◽  
Minkowski Sum ◽  
Duality Property ◽  
Finite Set ◽  
Minkowski Sums

This article proposes the use of polytopes in 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 poly-topes are useful for computing other operations as Minkowski sums. Then, this paper proposes a truncation algorithm to obtain the V-description of polytopes in ℝn from its 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 HV-polytopes is illustrated by the tolerance analysis of a real industrial case using the open source software PolitoCAT and politopix.

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