Minkowski sum
E14947
The Minkowski sum is a fundamental operation in geometry and convex analysis that combines two sets by adding every vector in one set to every vector in the other, widely used in areas such as optimization, robotics, and computational geometry.
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
binary operation on sets
→
concept in convex analysis → geometric operation → |
| alsoKnownAs |
vector sum of sets
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|
| appliesTo |
convex sets
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non-convex sets → |
| belongsTo |
convex geometry
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|
| definedOn |
subsets of Euclidean space
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subsets of a vector space → |
| distributesOver |
scalar multiplication of sets
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|
| field |
computational geometry
→
convex analysis → geometry → mathematical morphology → optimization → robotics → |
| generalizationOf |
addition of vectors
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|
| hasDefinition |
A + B = { a + b | a ∈ A, b ∈ B }
→
|
| hasHistoricalOrigin |
early 20th century
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|
| hasIdentityElement |
set containing only the zero vector
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|
| hasProperty |
distributes over finite unions
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is compatible with linear transformations → sum of closed sets is closed → sum of compact sets is compact → sum of polytopes is a polytope → sum of two convex sets is convex → |
| isAssociative |
true
→
|
| isCommutative |
true
→
|
| isMonotone |
true
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|
| isTranslationInvariant |
true
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|
| namedAfter |
Hermann Minkowski
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|
| preservesBoundedness |
true
→
|
| preservesClosedness |
true
→
|
| preservesConvexity |
true
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|
| relatedConcept |
Minkowski difference
→
Minkowski functional → convex hull → support function → |
| usedFor |
computing reachability regions
→
shape offsetting → summing random sets in stochastic geometry → tolerance analysis in CAD → |
| usedIn |
collision detection
→
configuration space obstacles in robotics → linear programming geometry → morphological dilation → motion planning → polyhedral computations → support function calculations → |
Referenced by (2)
| Subject (surface form when different) | Predicate |
|---|---|
|
Hermann Minkowski
→
Hermann Minkowski → |
knownFor |