- Example
- Convex hulls of Minkowski sums
- Lp Minkowski sum
- See also
- Notes
- References
- External links
In geometry, the Minkowski sum (also known as dilation) of two sets of position vectors A and B in Euclidean space is formed by adding each vector in A to each vector in B, i.e., the set
Analogously, the Minkowski difference (or geometric difference)[1] is defined as
It is important to note that in general 
. For instance, in a one-dimensional case 
and 
the Minkowski difference 
, whereas 
The correct formula connecting Minkowski sum and difference is as follows (here 
stands for the complement of 
):
In a two-dimensional case, Minkowski difference is closely related to erosion (morphology) in image processing.
The concept is named for Hermann Minkowski.
Example
For example, if we have two sets A and B, each consisting of three position vectors (informally, three points), representing the vertices of two triangles in 
, with coordinates
and
then their Minkowski sum is
which comprises the vertices of a hexagon.
For Minkowski addition, the zero set, {0}, containing only the zero vector, 0, is an identity element: for every subset S of a vector space,
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:
Convex hulls of Minkowski sums
Minkowski addition behaves well with respect to the operation of taking convex hulls, as shown by the following proposition:
- For all non-empty subsets S1 and S2 of a real vector space, the convex hull of their Minkowski sum is the Minkowski sum of their convex hulls:
This result holds more generally for any finite collection of non-empty sets:
In mathematical terminology, the operations of Minkowski summation and of forming convex hulls are commuting operations.[2][3]
If {{mvar|S}} is a convex set then 
is also a convex set; furthermore
for every 
. Conversely, if this "distributive property" holds for all non-negative real numbers, 
, then the set is convex.[4]
The figure shows an example of a non-convex set for which {{math|A + A ⊋ 2A}}|title=General competitive analysis|publisher=North-Holland||series=Advanced textbooks in economics|volume=12|edition=reprint of (1971) San Francisco, CA: Holden-Day, Inc. Mathematical economics texts. 6|location=Amsterdam|isbn=978-0-444-85497-1|mr=439057|ref=harv}}
|author=Henry Mann |authorlink=Henry Mann
|title=Addition Theorems: The Addition Theorems of Group Theory and Number Theory
|publisher=Robert E. Krieger Publishing Company
|via=http://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. 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=978-0-691-01586-6|mr=1451876|ref=harv}}
- {{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 |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 }}.
- {{Citation |first=Rolf |last=Schneider |title=Convex bodies: the Brunn-Minkowski theory |publisher=Cambridge University Press |location=Cambridge |year=1993 |isbn= }}.
- {{Citation |first=Terence |last=Tao |lastauthoramp=yes |first2=Van |last2=Vu |title=Additive Combinatorics |publisher=Cambridge University Press |year=2006 |isbn= }}.
- {{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|authorlink=Roger Evans Howe|last=Howe|first=Roger|year=1979|publisher=Cowles Foundation for Research in Economics, Yale University|series=Cowles Foundation discussion papers|volume=538|url=http://econpapers.repec.org/RePEc:cwl:cwldpp:538}}
- Minkowski Sums, in Computational Geometry Algorithms Library
- The Minkowski Sum of Two Triangles and The Minkowski Sum of a Disk and a Polygon by George Beck, The Wolfram Demonstrations Project.
- Minkowski's addition of convex shapes by Alexander Bogomolny: an applet
- OpenSCAD User Manual/Transformations#minkowski by Marius Kintel: Application
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