View source for Minkowski addition

{{short description|Sums vector sets A and B by adding each vector in A to each vector in B}}
[[File:Сумма Минковского.svg|thumb|alt=|The red figure is the Minkowski sum of blue and green figures.]]

In [[geometry]], the '''Minkowski sum''' of two [[set (mathematics)|sets]] of [[position vector]]s ''A'' and ''B'' in [[Euclidean space]] is formed by [[vector addition|adding each vector]] in ''A'' to each vector in ''B'':

<math display="block">A + B = \{\mathbf{a}+\mathbf{b}\,|\,\mathbf{a}\in A,\ \mathbf{b}\in B\}</math>

The '''Minkowski difference''' (also ''Minkowski subtraction'', ''Minkowski decomposition'', or ''geometric difference'')<ref>{{citation |last=Hadwiger |first=Hugo|title=Minkowskische Addition und Subtraktion beliebiger Punktmengen und die Theoreme von Erhard Schmidt |journal=Mathematische Zeitschrift |volume=53 |issue=3 |pages=210–218 |year=1950 |doi=10.1007/BF01175656 |s2cid=121604732 |url=https://gdz.sub.uni-goettingen.de/id/PPN266833020_0053?tify={%22pages%22:[214]} |access-date=2023-01-12}}</ref> is the corresponding inverse, where <math display="inline">(A - B)</math> produces a set that could be summed with ''B'' to recover ''A''. This is defined as the [[Complement (set theory)|complement]] of the Minkowski sum of the complement of ''A'' with the reflection of ''B'' about the origin.<ref>{{cite thesis |last=Li |first=Wei |date=Fall 2011 |title=GPU-Based Computation of Voxelized Minkowski Sums with Applications |url=https://escholarship.org/uc/item/9rm7j1pq |type=PhD |publisher=[[UC Berkeley]] |pages=13–14 |access-date=2023-01-10}}</ref>

<math display="block">\begin{align}
-B &= \{\mathbf{-b}\,|\,\mathbf{b}\in B\}\\
A - B &= (A^\complement + (-B))^\complement
\end{align}</math>

This definition allows a symmetrical relationship between the Minkowski sum and difference. Note that alternately taking the sum and difference with ''B'' is not necessarily equivalent. The sum can fill gaps which the difference may not re-open, and the difference can erase small islands which the sum cannot recreate from nothing.

<math display="block">\begin{align}
(A - B) + B &\subseteq A\\
(A + B) - B &\supseteq A\\
A - B &= (A^\complement + (-B))^\complement\\
A + B &= (A^\complement - (-B))^\complement\\
\end{align}</math>

In 2D [[image processing]] the Minkowski sum and difference are known as [[Dilation (morphology)|dilation]] and [[Erosion (morphology)|erosion]].

An alternative definition of the Minkowski difference is sometimes used for computing intersection of convex shapes.<ref>{{cite journal |last=Lozano-Pérez |first=Tomás |date=February 1983 |title=Spatial Planning: A Configuration Space Approach |url=https://lis.csail.mit.edu/pubs/tlp/spatial-planning.pdf |journal=[[IEEE Transactions on Computers]] |volume=C-32 |issue=2 |pages=111 |doi=10.1109/TC.1983.1676196 |hdl=1721.1/5684 |s2cid=18978404 |access-date=2023-01-10}}</ref> This is not equivalent to the previous definition, and is not an inverse of the sum operation. Instead it replaces the vector addition of the Minkowski sum with a [[vector subtraction]]. If the two convex shapes intersect, the resulting set will contain the origin.

<math display="block">A - B = \{\mathbf{a}-\mathbf{b}\,|\,\mathbf{a}\in A,\ \mathbf{b}\in B\} = A + (-B)</math>

The concept is named for [[Hermann Minkowski]].

== Example ==
[[File:Minkowski-sumex4.svg|thumb|Minkowski sum {{nowrap|''A'' + ''B''}}]]
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 display="inline">\mathbb{R}^2</math>, with coordinates

<math display="block">A = \{(1,0), (0,1), (0,-1)\}</math>

and

<math display="block">B = \{(0,0), (1,1), (1,-1)\}</math>

then their Minkowski sum is

<math display="block">A + B = \{(1,0), (2,1), (2,-1), (0,1), (1,2), (1,0), (0,-1), (1,0), (1,-2)\},</math>

which comprises the vertices of a hexagon and its center.

For Minkowski addition, the {{em|zero set}}, <math display="inline">\{ 0 \},</math> containing only the [[zero vector]], 0, is an [[identity element]]: for every subset ''S'' of a vector space,

<math display="block">S + \{0\} = S.</math>

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 display="block">S + \emptyset = \emptyset.</math>

For another example, consider the Minkowski sums of open or closed balls in the field <math display="inline">\mathbb{K},</math> which is either the [[real number]]s <math display="inline">\R</math> or [[complex number]]s <math display="inline">\C</math>. If <math display="inline">B_r:= \{ s \in \mathbb{K}: |s| \leq r \}</math> is the closed ball of radius <math display="inline">r \in [0, \infty]</math> centered at <math display="inline">0</math> in <math display="inline">\mathbb{K}</math> then for any <math display="inline">r, s \in [0, \infty]</math>, <math display="inline">B_r + B_s = B_{r+s}</math> and also <math display="inline">c B_r = B_{|c|r}</math> will hold for any scalar <math display="inline">c \in \mathbb{K}</math> such that the product <math display="inline">|c|r</math> is defined (which happens when <math display="inline">c \neq 0</math> or <math display="inline">r \neq \infty</math>). If <math display="inline">r</math>, <math display="inline">s</math>, and <math display="inline">c</math> are all non-zero then the same equalities would still hold had <math display="inline">B_r</math> been defined to be the open ball, rather than the closed ball, centered at 0 (the non-zero assumption is needed because the open ball of radius 0 is the empty set). The Minkowski sum of a closed ball and an open ball is an open ball. More generally, the Minkowski sum of an [[Open set|open subset]] with {{em|any}} other set will be an open subset.

If <math display="inline">G = \{ (x, 1/x) : 0 \neq x \in \R \}</math> is the [[Graph of a function|graph]] of <math display="inline">f(x) = \frac{1}{x}</math> and if and <math display="inline">Y = \{ 0 \} \times \R</math> is the <math display="inline">y</math>-axis in <math display="inline">X = \R^2</math> then the Minkowski sum of these two [[Closed set|closed subsets]] of the plane is the [[open set]] <math display="inline">G + Y = \{ (x, y) \in \R^2 : x \neq 0 \} = \R^2 \setminus Y</math> consisting of everything other than the <math display="inline">y</math>-axis. This shows that the Minkowski sum of two [[closed set]]s is not necessarily a closed set. However, the Minkowski sum of two closed subsets will be a closed subset if at least one of these sets is also a [[Compact space|compact subset]].

== Convex hulls of Minkowski sums ==
Minkowski addition behaves well with respect to the operation of taking [[convex hull]]s, as shown by the following proposition:

{{block indent| For all non-empty subsets <math display="inline">S_1</math> and <math display="inline">S_2</math> of a real vector space, the convex hull of their Minkowski sum is the Minkowski sum of their convex hulls:

<math display="block">\operatorname{Conv}(S_1 + S_2) = \operatorname{Conv}(S_1) + \operatorname{Conv}(S_2).</math>}}

This result holds more generally for any finite collection of non-empty sets:

<math display="block">\operatorname{Conv}\left(\sum{S_n}\right) = \sum\operatorname{Conv}(S_n).</math>

In mathematical terminology, the [[Operation (mathematics)|operation]]s of Minkowski summation and of forming [[convex hull]]s are [[commutativity|commuting]] operations.<ref>Theorem&nbsp;3 (pages&nbsp;562–563): {{cite journal|first1=M.|last1=Krein|author-link1=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 |series=Second Series|volume=41|issue=3 |pages=556–583|doi=10.2307/1968735|mr=2009 | jstor = 1968735}}</ref><ref>For the commutativity of Minkowski addition and [[Convex hull|convexification]], see Theorem&nbsp;1.1.2 (pages&nbsp;2–3) in Schneider; this reference discusses much of the literature on the [[convex hull]]s of Minkowski [[sumset]]s in its "Chapter&nbsp;3 Minkowski addition" (pages&nbsp;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&nbsp;University Press|location=Cambridge|year=1993|pages=xiv+490|isbn=978-0-521-35220-8|mr=1216521|url=https://archive.org/details/convexbodiesbrun0000schn}}</ref>

If <math display="inline">S</math> is a convex set then <math>\mu S + \lambda S</math> is also a convex set; furthermore

<math display="block">\mu S + \lambda S = (\mu + \lambda)S</math>

for every <math display="inline">\mu,\lambda \geq 0</math>. Conversely, if this "[[distributive property]]" holds for all non-negative real numbers, <math display="inline">\mu</math> and <math display="inline">\lambda</math>, then the set is convex.<ref>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&nbsp;University Press|location=Cambridge|year=1993|pages=xiv+490|isbn=978-0-521-35220-8|mr=1216521|url=https://archive.org/details/convexbodiesbrun0000schn}}</ref>

[[File:Minkowskisum.svg|thumb|An example of a non-convex set such that <math display="inline">A + A \neq 2 A.</math>]]
The figure to the right shows an example of a non-convex set for which <math display="inline">A + A \subsetneq 2 A.</math>

An example in one dimension is: <math display="inline">B = [1, 2] \cup [4, 5].</math> It can be easily calculated that <math display="inline">2 B = [2, 4] \cup [8, 10]</math> but <math display="inline">B + B = [2, 4] \cup [5, 7] \cup [8, 10],</math> hence again <math display="inline">B + B \subsetneq 2 B.</math>

Minkowski sums act linearly on the perimeter of two-dimensional convex bodies: the perimeter of the sum equals the sum of perimeters. Additionally, if <math display="inline">K</math> is (the interior of) a [[curve of constant width]], then the Minkowski sum of <math display="inline">K</math> 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>

== Applications ==
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]]. It has also been shown to be closely connected to the [[Earth mover's distance]], and by extension, [[Transportation theory (mathematics)|optimal transport]].<ref>{{cite journal | journal = Discrete Applied Mathematics | title = Properties of the d-dimensional earth mover's problem | last1 = Kline | first1 = Jeffery | volume = 265 | year = 2019 | pages = 128–141 | doi = 10.1016/j.dam.2019.02.042| s2cid = 127962240 | doi-access = free }}</ref>

=== Motion planning ===
Minkowski sums are used in [[motion planning]] of an object among obstacles. They are used for the computation of the [[Configuration space (physics)|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.

=== Numerical control (NC) machining ===
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.

=== 3D solid modeling ===
In [[OpenSCAD]] Minkowski sums are used to outline a shape with another shape creating a composite of both shapes.

=== Aggregation theory ===
Minkowski sums are also frequently used in aggregation theory when individual objects to be aggregated are characterized via sets.<ref>{{cite journal | last1 = Zelenyuk | first1 = V. | year = 2015 | title = Aggregation of scale efficiency | url = https://ideas.repec.org/a/eee/ejores/v240y2015i1p269-277.html | journal = European Journal of Operational Research | volume = 240 | issue = 1| pages = 269–277 | doi=10.1016/j.ejor.2014.06.038}}</ref><ref>{{cite journal | last1 = Mayer | first1 = A. | last2 = Zelenyuk | first2 = V. | year = 2014 | title = Aggregation of Malmquist productivity indexes allowing for reallocation of resources | url = https://ideas.repec.org/a/eee/ejores/v238y2014i3p774-785.html | journal = European Journal of Operational Research | volume = 238 | issue = 3| pages = 774–785 | doi=10.1016/j.ejor.2014.04.003}}</ref>

=== Collision detection ===
Minkowski sums, specifically Minkowski differences, are often used alongside [[Gilbert–Johnson–Keerthi distance algorithm|GJK algorithms]] to compute [[collision detection]] for convex hulls in [[physics engines]].

== Algorithms for computing Minkowski sums ==
[[File:Shapley–Folkman lemma.svg|thumb|upright=1.4| 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.
| 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.
]]

=== Planar case ===
==== Two convex polygons in the plane ====
For two [[convex polygon]]s {{var|P}} and {{var|Q}} in the plane with {{var|m}} and {{var|n}} vertices, their Minkowski sum is a convex polygon with at most {{var|m}} + {{var|n}} vertices and may be computed in time O({{var|m}} + {{var|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 {{var|P}} and {{var|Q}} into a single ordered sequence {{var|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 {{var|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 {{var|P}} and {{var|Q}}.

==== Other ====
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>).

== Essential Minkowski sum ==
There is also a notion of the '''essential Minkowski sum''' +<sub>e</sub> of two subsets of Euclidean space. The usual Minkowski sum can be written as

<math display="block">A + B = \left\{ z \in \mathbb{R}^{n} \,|\, A \cap (z - B) \neq \emptyset \right\}.</math>

Thus, the '''essential Minkowski sum''' is defined by

<math display="block">A +_{\mathrm{e}} B = \left\{ z \in \mathbb{R}^{n} \,|\, \mu \left[A \cap (z - B)\right] > 0 \right\},</math>

where ''μ'' denotes the ''n''-dimensional [[Lebesgue measure]]. The reason for the term "essential" is the following property of [[indicator function]]s: while

<math display="block">1_{A \,+\, B} (z) = \sup_{x \,\in\, \mathbb{R}^{n}} 1_{A} (x) 1_{B} (z - x),</math>

it can be seen that

<math display="block">1_{A \,+_{\mathrm{e}}\, B} (z) = \mathop{\mathrm{ess\,sup}}_{x \,\in\, \mathbb{R}^{n}} 1_{A} (x) 1_{B} (z - x),</math>

where "ess&thinsp;sup" denotes the [[essential supremum]].

== ''L<sup>p</sup>'' Minkowski sum ==
For ''K'' and ''L'' compact convex subsets in <math display="inline">\mathbb{R}^n</math>, the Minkowski sum can be described by the [[support function]] of the convex sets:

<math display="block">h_{K+L} = h_K + h_L.</math>

For ''p'' ≥ 1, Firey<ref>{{citation|last=Firey|first=William J.|title=''p''-means of convex bodies|journal=Mathematica Scandinavica|volume=10|pages=17–24|year=1962|doi=10.7146/math.scand.a-10510|doi-access=free}}</ref> defined the '''''L''<sup>''p''</sup> Minkowski sum''' {{nowrap|''K'' +{{sub|''p''}} ''L''}} of compact convex sets ''K'' and ''L'' in <math>\mathbb{R}^n</math> containing the origin as

<math display="block">h_{K +_p L}^p = h_K^p + h_L^p.</math>

By the [[Minkowski inequality]], the function ''h{{sub|K+{{sub|p}}L}}'' is again positive homogeneous and convex and hence the support function of a compact convex set. This definition is fundamental in the ''L''<sup>''p''</sup> Brunn-Minkowski theory.

== See also ==
* {{annotated link|Blaschke sum}}
* {{annotated link|Brunn–Minkowski theorem}}, an inequality on the volumes of Minkowski sums
* {{annotated link|Convolution}}
* {{annotated link|Dilation (morphology)|Dilation}}
* {{annotated link|Erosion (morphology)|Erosion}}
* {{annotated link|Interval arithmetic}}
* {{annotated link|Mixed volume}} (a.k.a. [[Quermassintegral]] or [[intrinsic volume]])
* {{annotated link|Parallel curve}}
* {{annotated link|Shapley–Folkman lemma}}
* {{annotated link|Sumset}}
* {{annotated link|Topological vector space#Properties}}
* {{annotated link|Zonotope}}

== Notes ==
{{reflist}}

== References ==
* {{cite book|last1=Arrow|first1=Kenneth&nbsp;J.|author-link1=Kenneth Arrow|last2=Hahn|first2=Frank&nbsp;H.|author-link2=Frank Hahn|year=1980<!-- |chapter=Appendix&nbsp;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&nbsp;Francisco,&nbsp;CA: Holden-Day,&nbsp;Inc. Mathematical economics texts.&nbsp;'''6'''|location=Amsterdam|isbn=978-0-444-85497-1|mr=439057}}
* {{citation |last=Gardner |first=Richard J. |title=The Brunn-Minkowski inequality |journal=Bull. Amer. Math. Soc. (N.S.) | volume=39 | issue=3 | year=2002 | pages=355–405 (electronic) | doi=10.1090/S0273-0979-02-00941-2 | doi-access=free }}
* {{cite book|first1=Jerry|last1=Green|first2=Walter&nbsp;P.|last2=Heller|chapter=1 Mathematical&nbsp;analysis and&nbsp;convexity with applications to economics|pages=15–52|doi=10.1016/S1573-4382(81)01005-9|title=Handbook of mathematical&nbsp;economics, Volume&nbsp;'''I'''|editor1-link=Kenneth Arrow |editor1-first=Kenneth&nbsp;Joseph|editor1-last=Arrow|editor2-first=Michael&nbsp;D<!--. -->|editor2-last=Intriligator|series=Handbooks in economics|volume=1|publisher=North-Holland Publishing&nbsp;Co|location=Amsterdam|year=1981|isbn=978-0-444-86126-9|mr=634800}}
* {{citation
 |author=Henry Mann |author-link=Henry Mann
 |title=Addition Theorems: The Addition Theorems of Group Theory and Number Theory
 |publisher=Robert E. Krieger Publishing Company
 |via=www.krieger-publishing.com/subcats/MathematicsandStatistics/mathematicsandstatistics.html
 |location=Huntington, New York
 |year=1976
 |edition=Corrected reprint of 1965 Wiley
 |isbn=978-0-88275-418-5
 }}
* {{cite book|last=Rockafellar|first=R.&nbsp;Tyrrell|author-link=R. Tyrrell Rockafellar|title=Convex analysis|edition=Reprint of the 1979 Princeton mathematical series&nbsp;'''28'''|series=Princeton landmarks in mathematics|publisher=Princeton University Press|location=Princeton, NJ|year=1997|pages=xviii+451|isbn=978-0-691-01586-6|mr=1451876}}
* {{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 }}.
* {{citation |last1=Oks |first1=Eduard |last2=Sharir |first2=Micha | authorlink2=Micha Sharir |year=2006 |title=Minkowski Sums of Monotone and General Simple Polygons |journal=[[Discrete & Computational Geometry]] |volume=35 |issue=2 |pages=223–240 |doi=10.1007/s00454-005-1206-y |doi-access=free }}.
* {{citation |first=Rolf |last=Schneider |title=Convex bodies: the Brunn-Minkowski theory |publisher=Cambridge University Press |location=Cambridge |year=1993 }}.
* {{citation |first1=Terence |last1=Tao |name-list-style=amp |first2=Van |last2=Vu |title=Additive Combinatorics |publisher=Cambridge University Press |year=2006 }}.
* {{cite journal | last1 = Mayer | first1 = A. | last2 = Zelenyuk | first2 = V. | year = 2014 | title = Aggregation of Malmquist productivity indexes allowing for reallocation of resources | url = https://ideas.repec.org/a/eee/ejores/v238y2014i3p774-785.html | journal = European Journal of Operational Research | volume = 238 | issue = 3| pages = 774–785 | doi=10.1016/j.ejor.2014.04.003}}
* {{cite journal | last1 = Zelenyuk | first1 = V | year = 2015 | title = Aggregation of scale efficiency | url = https://ideas.repec.org/a/eee/ejores/v240y2015i1p269-277.html | journal = European Journal of Operational Research | volume = 240 | issue = 1| pages = 269–277 | doi=10.1016/j.ejor.2014.06.038}}

== External links ==
* {{springer|title=Minkowski addition|id=p/m120210}}
* {{citation|title=On the tendency toward convexity of the vector sum of sets|author-link=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}}
* [http://www.cgal.org/Pkg/MinkowskiSum2 Minkowski Sums], in [[Computational Geometry Algorithms Library]]
* [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]].
* [http://www.cut-the-knot.org/Curriculum/Geometry/PolyAddition.shtml Minkowski's addition of convex shapes] by [[Alexander Bogomolny]]: an applet
* [[Wikibooks:OpenSCAD User Manual/Transformations#minkowski]] by Marius Kintel: Application
* [https://minkowski-sum.herokuapp.com/minkownski-sum.html Application of Minkowski Addition to robotics] by Joan Gerard
* [https://github.com/jeff-kline/GEM Demonstration of Minkowski additivity, convex monotonicity, and other properties of the Earth Movers distance]

{{Convex analysis and variational analysis}}
{{Functional analysis}}
{{Topological vector spaces}}

[[Category:Abelian group theory]]
[[Category:Affine geometry]]
[[Category:Binary operations]]
[[Category:Convex geometry]]
[[Category:Digital geometry]]
[[Category:Geometric algorithms]]
[[Category:Hermann Minkowski]]
[[Category:Sumsets]]
[[Category:Theorems in convex geometry]]
[[Category:Variational analysis]]