Minkowski functional
E14948
The Minkowski functional is a mathematical tool in functional analysis that assigns a nonnegative real number to each vector in a vector space based on its position relative to a given convex, balanced, absorbing set, generalizing the notion of a norm.
Statements (44)
| Predicate | Object |
|---|---|
| instanceOf |
functional
→
mathematical concept → |
| alsoKnownAs |
gauge
→
gauge functional → |
| application |
characterizing bounded sets in locally convex spaces
→
defining metrics on topological vector spaces → formulating Hahn–Banach type results → |
| associatesToEachSet |
a sublinear functional
→
|
| category |
functional-analytic construction
→
tool in convex geometry → |
| codomain |
nonnegative real numbers
→
|
| conditionForNorm |
set is absorbing
→
set is balanced → set is bounded → set is closed → set is convex → set is symmetric about the origin → |
| definedUsing |
absorbing set
→
balanced set → convex set → |
| dependsOn |
choice of convex balanced absorbing set
→
|
| domain |
vector space
→
|
| field |
convex analysis
→
functional analysis → |
| generalizes |
norm
→
|
| input |
vector
→
|
| invariantUnder |
scaling of the defining set by positive constants up to equivalence
→
|
| namedAfter |
Hermann Minkowski
→
|
| output |
nonnegative scalar
→
|
| property |
nonnegative
→
positively homogeneous → subadditive → sublinear → vanishes at the origin → |
| relatedTo |
gauge of a convex set
→
seminorm → support function → |
| typicalAssumptionOnSet |
contains the origin
→
|
| usedIn |
construction of locally convex topologies
→
definition of seminorms → duality theory in convex analysis → study of convex bodies → theory of locally convex spaces → |
| yields |
norm on the linear span of the set under suitable conditions
→
|
Referenced by (3)
| Subject (surface form when different) | Predicate |
|---|---|
|
Hermann Minkowski
→
Hermann Minkowski → |
knownFor |
|
Minkowski sum
→
|
relatedConcept |