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| [[File:Сумма Минковского.svg|thumb|alt=|The red figure is the Minkowski sum of blue and green figures.]]
| | Offer a strategy and often battle activation where you must manage your man or women tribe and also protect it. You have so as to build constructions which is likely to provide protection for all your soldiers along with our instruction. First target on your protection together with after its recently been taken treatment. You might need to move forward which has the criminal offense product. As well as your Military facilities, you in addition need to keep in mind the way your group is certainly going. For instance, collecting indicates as well as raising your own tribe is the key to good leads.<br><br> |
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| In [[geometry]], the '''Minkowski sum''' (also known as [[Dilation (morphology)|dilation]]) of two [[set (mathematics)|sets]] of [[position vector]]s ''A'' and ''B'' in [[Euclidean space]] is formed by adding each vector in ''A'' to each vector in ''B'', i.e. the set
| | If you are purchasing a game your child, appear for an individual that allows several individuals to do together. Gaming might just be a singular activity. If you have any questions concerning where and how to use clash of clans cheats ([http://circuspartypanama.com relevant internet site]), you can get in touch with us at our own page. Nonetheless, it's important to service your [http://www.Alexa.com/search?q=youngster&r=topsites_index&p=bigtop youngster] to continually be societal, and multiplayer clash of clans trucos gaming can do that. They allow siblings as well buddies to all sit a while and laugh and strive to compete together.<br><br>Nevertheless, if you want avoid at the top of one's competitors, there are several simple points you need to keep in mind. Realize your foe, grasp the game and the success will be yours. It is possible to take the aid of clash of clans hack tools and some other rights if you like in your course. Absolutely for your convenience, allow me to share the general details in this particular sport that you must have to remember of. Saw all of them carefully!<br><br>Guilds and clans have ended up popular ever since the particular beginning of first-person gift shooter and MMORPG gambling. World of WarCraft develops to the concept with their own World associated [http://www.wonderhowto.com/search/Warcraft+guilds/ Warcraft guilds]. A real guild can easily always try to be understood as a in respect of players that band back down for companionship. Folks the guild travel back together again for fun and excit while improving in ordeal and gold.<br><br>No matter the reason, computer game hacks are widespread and spread fairly rapidly over the online market place. The gaming community is trying to find means stay away from cheaters from overrunning regarding game; having lots related with cheaters playing a one particular game can really produce honest players to naturally quit playing, or play only with friends they trust. This poses a extensive problem particularly for price games for example EverQuest, wherein a loss in players ultimately result within a loss of income.<br><br>Ought to you perform online multi-player game titles, don't skip the strength of mood of voice chat! A mic or bluetooth headset is a very simple expenditure, and having our capability to speak to successfully your fellow athletes makes a lot of features. You are within a position to create more valuable connections with the egaming community and stay an actual far more successful party person when you were able connect out loud.<br><br>Pc games or computer games have increased in popularity nowadays, not with the younger generation, but also with uncle and aunts as well. There are a number games available, ranging from the intellectual to the each day - your options can be found limitless. Online position playing games are one of the most popular games anywhere remaining. With this popularity, plenty of men and women are exploring and trying to find ways to go the actual whole game as very fast as they can; reasons for using computer How to compromise in clash of clans range from simply trying to own your own friends stare at you living in awe, or getting whole lot of game money anyone really can sell later, or simply just to rid the game among the fun factor for the additional players. |
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| : <math>A + B = \{\mathbf{a}+\mathbf{b}\,|\,\mathbf{a}\in A,\ \mathbf{b}\in B\}.</math>
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| Analogously, the '''Minkowski difference''' is defined as
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| : <math>A - B = \{\mathbf{a}-\mathbf{b}\,|\,\mathbf{a}\in A,\ \mathbf{b}\in B\}.</math>
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| [[File:Minkowski-sumex4.svg|thumb|Minkowski sum {{math|''A'' + ''B''}}]]
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| [[File:Minkowski-sumex2.svg|thumb|''B'']]
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| [[File:Minkowski-sumex1.svg|thumb|''A'']]
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| ==Example==
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| For example, if we have two sets ''A'' and ''B'', each consisting of three position vectors (informally, three points), representing the [[vertex (geometry)|vertices]] of two ([[triangle]]s) in <math>\mathbb{R}^2</math>, with coordinates
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| :{{math|1=''A'' = {(1, 0), (0, 1), (0, −1)} }}
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| and
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| :{{math|1=''B'' = {(0, 0), (1, 1), (1, −1)} }},
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| then the Minkowski sum is<br/>
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| {{math|1=''A'' + ''B'' = {{{math|(1, 0)}}, {{math|(2, 1)}}, {{math|(2, −1)}}, {{math|(0, 1)}}, {{math|(1, 2)}}, {{math|(1, 0)}}, {{math|(0, −1)}}, {{math|(1, 0)}}, {{math|(1, −2)}}} }}, which looks like a [[hexagon]], with three 'repeated' points at {{math|(1, 0)}}.
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| For Minkowski addition, the ''zero set'' {0}, containing only the [[zero vector]] 0, is an [[identity element]]: For every<!-- why only non-empty? for empty S the result will be empty too, i.e. identical --> subset S, of a vector space
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| : S + {0} = S;
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| The [[empty set]] is important in Minkowski addition, because the empty set annihilates every other subset: for every subset, S, of a vector space, its sum with the empty set is empty: {{math|1=S + <math>\emptyset</math> = <math>\emptyset</math>}}.
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| [[File:Extreme points illustration.png|thumb|right|alt=A picture of a smoothed triangle, like a triangular tortilla-chip or a triangular road-sign. Each of the three rounded corners is drawn with a red curve. The remaining interior points of the triangular shape are shaded with blue.|In the [[convex hull]] of the red set, each blue point is a [[convex combination]] of some red points.]] | |
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| [[File:Minkowski sum.png|thumb|alt=Three squares are shown in the non-negative quadrant of the Cartesian plane. The square Q<sub>1</sub>={{closed-closed|0,1}}×{{closed-closed|0,1}} is green. The square Q<sub>2</sub>={{closed-closed|1,2}}×{{closed-closed|1,2}} is brown, and it sits inside the turquoise square Q<sub>1</sub>+Q<sub>2</sub>={{closed-closed|1,3}}×{{closed-closed|1,3}}.|'''Minkowski addition''' of sets. The <!-- '''Minkowski''' -->sum of the squares ''Q''<sub>1</sub>={{closed-closed|0,1}}<sup>2</sup> and ''Q''<sub>2</sub>={{closed-closed|1,2}}<sup>2</sup> is the square ''Q''<sub>1</sub>+''Q''<sub>2</sub>={{closed-closed|1,3}}<sup>2</sup>.]]
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| ==Convex hulls of Minkowski sums==
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| Minkowski addition behaves well with respect to the operation of taking [[convex hull]]s, as shown by the following proposition:
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| * For all subsets S<sub>1</sub> and S<sub>2</sub> of a real vector-space, the convex hull of their Minkowski sum is the Minkowski sum of their convex hulls
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| : Conv(S<sub>1</sub> + S<sub>2</sub>) = Conv(S<sub>1</sub>) + Conv(S<sub>2</sub>).
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| This result holds more generally for each finite collection of non-empty sets
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| : Conv(∑S<sub>n</sub>) = ∑Conv(S<sub>n</sub>).
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| In mathematical terminology, the [[operation (mathematics)|operation]]s of Minkowski summation and of forming [[convex hull]]s are [[commutativity|commuting]] operations.<ref>Theorem 3 (pages 562–563): {{cite news|first1=M.|last1=Krein|authorlink1=Mark Krein|first2=V.|last2=Šmulian|year=1940|title=On regularly convex sets in the space conjugate to a Banach space|journal=Annals of Mathematics (2), Second series|volume=41|pages=556–583|doi=10.2307/1968735|mr=2009 | jstor = 1968735}}</ref><ref name="Schneider">For the commutativity of [[Minkowski sum|Minkowski addition]] and [[convex hull|convexification]], see Theorem 1.1.2 (pages 2–3) in Schneider; this reference discusses much of the literature on the [[convex hull]]s of Minkowski [[sumset]]s in its "Chapter 3 Minkowski addition" (pages 126–196): {{cite book|last=Schneider|first=Rolf|title=Convex bodies: The Brunn–Minkowski theory|series=Encyclopedia of mathematics and its applications|volume=44|publisher=Cambridge University Press|location=Cambridge|year=1993|pages=xiv+490 |isbn=0-521-35220-7|mr=1216521}}</ref>
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| If {{mvar|S}} is a convex set then also <math>\mu S+\lambda S</math> is a convex set; furthermore
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| :<math>\mu S+\lambda S=(\mu+\lambda)S</math> for every <math>\mu,\lambda \geq 0</math>.
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| Conversely, if this "[[distributive property]]" holds for all non-negative real numbers, <math>\mu, \lambda</math>, then the set is convex.<ref name="Schneider">Chapter 1: {{cite book|last=Schneider|first=Rolf|title=Convex bodies: The Brunn–Minkowski theory|series=Encyclopedia of mathematics and its applications|volume=44|publisher=Cambridge University Press|location=Cambridge|year=1993|pages=xiv+490 |isbn=0-521-35220-7|mr=1216521}}</ref> The figure shows an example of an non-convex set for which {{math|''A'' + ''A'' ≠ 2''A''}}<!-- don't use <math> for that due to broken spacing in HTML -->.
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| [[File:Minkowskisum.svg|thumb|An example of a non-convex set such that {{math|''A'' + ''A'' ≠ 2''A''}}]]
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| Minkowski sums act linearly on the perimeter of two-dimensional convex bodies: the perimeter of the sum equals the sum of perimeters. Additionally, if ''K'' is (the interior of) a [[curve of constant width]], then the Minkowski sum of ''K'' and of its 180° rotation is a disk. These two facts can be combined to give a short proof of [[Barbier's theorem]] on the perimeter of curves of constant width.<ref>[http://www.cut-the-knot.org/ctk/Barbier.shtml The Theorem of Barbier (Java)] at [[cut-the-knot]].</ref>
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| ==Applications==
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| Minkowski addition plays a central role in [[mathematical morphology]]. It arises in the [[brush-and-stroke paradigm]] of [[2D computer graphics]] (with various uses, notably by [[Donald E. Knuth]] in [[Metafont]]), and as the [[solid sweep]] operation of [[3D computer graphics]].
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| ===Motion planning===
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| Minkowski sums are used in [[motion planning]] of an object among obstacles. They are used for the computation of the [[configuration space]], which is the set of all admissible positions of the object. In the simple model of translational motion of an object in the plane, where the position of an object may be uniquely specified by the position of a fixed point of this object, the configuration space are the Minkowski sum of the set of obstacles and the movable object placed at the origin and rotated 180 degrees.
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| ===Numerical control (NC) machining===
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| In [[numerical control]] machining, the programming of the NC tool exploits the fact that the Minkowski sum of the [[cutting piece]] with its trajectory gives the shape of the cut in the material.
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| ==Algorithms for computing Minkowski sums==
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| [[File:Shapley–Folkman lemma.svg|thumb|300px|alt=Minkowski addition of four line-segments. The left-hand pane displays four sets, which are displayed in a two-by-two array. Each of the sets contains exactly two points, which are displayed in red. In each set, the two points are joined by a pink line-segment, which is the convex hull of the original set. Each set has exactly one point that is indicated with a plus-symbol. In the top row of the two-by-two array, the plus-symbol lies in the interior of the line segment; in the bottom row, the plus-symbol coincides with one of the red-points. This completes the description of the left-hand pane of the diagram. The right-hand pane displays the Minkowski sum of the sets, which is the union of the sums having exactly one point from each summand-set; for the displayed sets, the sixteen sums are distinct points, which are displayed in red: The right-hand red sum-points are the sums of the left-hand red summand-points. The convex hull of the sixteen red-points is shaded in pink. In the pink interior of the right-hand sumset lies exactly one plus-symbol, which is the (unique) sum of the plus-symbols from the right-hand side. The right-hand plus-symbol is indeed the sum of the four plus-symbols from the left-hand sets, precisely two points from the original non-convex summand-sets and two points from the convex hulls of the remaining summand-sets.|
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| |Minkowski addition and convex hulls. The sixteen dark-red points (on the right) form the Minkowski sum of the four non-convex sets (on the left), each of which consists of a pair of red points. Their convex hulls (shaded pink) contain plus-signs (+): The right plus-sign is the sum of the left plus-signs.]]
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| ===Planar case===
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| ====Two convex polygons in the plane====
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| For two [[convex polygon]]s P and Q in the plane with m and n vertices, their Minkowski sum is a convex polygon with at most m + n vertices and may be computed in time O (m + n) by a very simple procedure, which may be informally described as follows. Assume that the edges of a polygon are given and the direction, say, counterclockwise, along the polygon boundary. Then it is easily seen that these edges of the convex polygon are ordered by [[polar coordinate system|polar angle]]. Let us [[Merge algorithm|merge the ordered sequences]] of the directed edges from P and Q into a single ordered sequence S. Imagine that these edges are solid [[arrow]]s which can be moved freely while keeping them parallel to their original direction. Assemble these arrows in the order of the sequence S by attaching the tail of the next arrow to the head of the previous arrow. It turns out that the resulting [[polygonal chain]] will in fact be a convex polygon which is the Minkowski sum of P and Q.
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| ====Other====
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| If one polygon is convex and another one is not, the complexity of their Minkowski sum is O(nm). If both of them are nonconvex, their Minkowski sum complexity is O((mn)<sup>2</sup>).
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| ==Essential Minkowski sum==
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| There is also a notion of the '''essential Minkowski sum''' +<sub>e</sub> of two subsets of Euclidean space. Note that the usual Minkowski sum can be written as
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| :<math>A + B = \{ z \in \mathbb{R}^{n} \,|\, A \cap (\{z\} - B) \neq \emptyset \}.</math>
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| Thus, the '''essential Minkowski sum''' is defined by
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| :<math>A +_{\mathrm{e}} B = \{ z \in \mathbb{R}^{n} \,|\, \mu \left[A \cap (\{z\} - B)\right] > 0 \},</math>
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| where ''μ'' denotes the ''n''-dimensional [[Lebesgue measure]]. The reason for the term "essential" is the following property of [[indicator function]]s: while
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| :<math>1_{A \,+\, B} (z) = \sup_{x \,\in\, \mathbb{R}^{n}} 1_{A} (x) 1_{B} (z - x),</math>
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| it can be seen that
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| :<math>1_{A \,+_{\mathrm{e}}\, B} (z) = \mathop{\mathrm{ess\,sup}}_{x \,\in\, \mathbb{R}^{n}} 1_{A} (x) 1_{B} (z - x),</math>
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| where "ess sup" denotes the [[essential supremum]].
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| ==See also==
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| * [[Dilation (morphology)|Dilation]]
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| * [[Erosion (morphology)|Erosion]]
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| * [[Interval arithmetic]]
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| * [[Mixed volume]] (a.k.a. [[Quermassintegral]] or [[intrinsic volume]])
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| * [[Parallel curve]]
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| * [[Shapley–Folkman lemma]]
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| * [[Zonotope]]
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| * [[Convolution]]
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| ==Notes==
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| <references/>
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| ==References==
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| * {{cite book|last1=Arrow|first1=Kenneth J.|authorlink1=Kenneth Arrow|last2=Hahn|first2=Frank H.|authorlink2=Frank Hahn|year=1980<!-- |chapter=Appendix B: Convex and related sets -->|title=General competitive analysis|publisher=North-Holland|<!-- pages=375–401 -->|series=Advanced textbooks in economics|volume=12|edition=reprint of (1971) San Francisco, CA: Holden-Day, Inc. Mathematical economics texts. '''6'''|location=Amsterdam|isbn=0-444-85497-5|mr=439057|ref=harv}}
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| *{{Citation |last=Gardner |first=Richard J. |title=The Brunn-Minkowski inequality |journal=[[Bulletin of the American Mathematical Society|Bull. Amer. Math. Soc.]] (N.S.) | volume=39 | issue=3 | year=2002 | pages=355–405 (electronic) | doi=10.1090/S0273-0979-02-00941-2 }}
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| * {{cite book|first1=Jerry|last1=Green|first2=Walter P.|last2=Heller|chapter=1 Mathematical analysis and convexity with applications to economics|pages=15–52|url=http://www.sciencedirect.com/science/article/B7P5Y-4FDF0FN-5/2/613440787037f7f62d65a05172503737|doi=10.1016/S1573-4382(81)01005-9|title=Handbook of mathematical economics, Volume '''I'''|editor1-link=Kenneth Arrow |editor1-first=Kenneth Joseph|editor1-last=Arrow|editor2-first=Michael D<!-- . -->|editor2-last=Intriligator|series=Handbooks in economics|volume=1|publisher=North-Holland Publishing Co|location=Amsterdam|year=1981|isbn=0-444-86126-2|mr=634800|ref=harv}}
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| *{{Citation
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| |author=Henry Mann |authorlink=Henry Mann
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| |title=Addition Theorems: The Addition Theorems of Group Theory and Number Theory
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| |publisher=[http://www.krieger-publishing.com/subcats/MathematicsandStatistics/mathematicsandstatistics.html Robert E. Krieger Publishing Company]
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| |location=Huntington, New York
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| |year=1976
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| |edition=Corrected reprint of 1965 Wiley
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| |isbn=0-88275-418-1
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| }}
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| * {{cite book|last=Rockafellar|first=R. Tyrrell|authorlink=R. Tyrrell Rockafellar|title=Convex analysis|edition=Reprint of the 1979 Princeton mathematical series '''28'''|series=Princeton landmarks in mathematics|publisher=Princeton University Press|location=Princeton, NJ|year=1997|pages=xviii+451|isbn=0-691-01586-4|mr=1451876|<!-- Rockafellar, the "bible" of convex analysis, according to Lemaréchal and Hiart-Urruty, gets two MRs because it's the best-->id={{MR|274683}}|ref=harv}}
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| *{{Citation |first=Melvyn B. |last=Nathanson |title=Additive Number Theory: Inverse Problems and Geometry of Sumsets |series=GTM |volume=165 |publisher=Springer |year=1996 |zbl=0859.11003 }}.
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| *{{Citation |last=Oks |first=Eduard |last2=Sharir |first2=Micha |year=2006 |title=Minkowski Sums of Monotone and General Simple Polygons |journal=Discrete and Computational Geometry |volume=35 |issue=2 |pages=223–240 |doi=10.1007/s00454-005-1206-y }}.
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| *{{Citation |first=Rolf |last=Schneider |title=Convex bodies: the Brunn-Minkowski theory |publisher=Cambridge University Press |location=Cambridge |year=1993 |isbn= }}.
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| *{{Citation |first=Terence |last=Tao |lastauthoramp=yes |first2=Van |last2=Vu |title=Additive Combinatorics |publisher=Cambridge University Press |year=2006 |isbn= }}.
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| ==External links==
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| * {{springer|title=Minkowski addition|id=p/m120210}}
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| * {{citation|title=On the tendency toward convexity of the vector sum of sets|authorlink=Roger Evans Howe|last=Howe|first=Roger|year=1979|publisher=[[Cowles Foundation|Cowles Foundation for Research in Economics]], Yale University|series=Cowles Foundation discussion papers|volume=538|url=http://econpapers.repec.org/RePEc:cwl:cwldpp:538}}
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| * [http://www.cgal.org/Pkg/MinkowskiSum2 Minkowski Sums], in [[Computational Geometry Algorithms Library]]
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| * [http://demonstrations.wolfram.com/TheMinkowskiSumOfTwoTriangles/ The Minkowski Sum of Two Triangles] and [http://demonstrations.wolfram.com/TheMinkowskiSumOfADiskAndAPolygon/ The Minkowski Sum of a Disk and a Polygon] by George Beck, [[The Wolfram Demonstrations Project]].
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| * [http://www.cut-the-knot.org/Curriculum/Geometry/PolyAddition.shtml Minkowski's addition of convex shapes] by Alexander Bogomolny: an applet
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| {{Functional Analysis}}
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| [[Category:Convex geometry]]
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| [[Category:Binary operations]]
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| [[Category:Digital geometry]]
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| [[Category:Geometric algorithms]]
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| [[Category:Sumsets]]
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| [[Category:Variational analysis]]
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| [[Category:Abelian group theory]]
| |
| [[Category:Affine geometry]]
| |
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