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- Im mathematischen Teilgebiet der Funktionalanalysis ist das Minkowski-Funktional (nach Hermann Minkowski), oft auch Eichfunktional genannt, eine Verallgemeinerung des Normbegriffes. (de)
- En géométrie, la notion de jauge généralise celle de semi-norme. À toute partie C d'un ℝ-espace vectoriel E on associe sa jauge, ou fonctionnelle de Minkowski pC, qui est une application de E dans [0, +∞] mesurant, pour chaque vecteur, par quel rapport il faut dilater C pour englober ce vecteur. Dès que C contient l'origine, pC est positivement homogène ; si C est étoilée par rapport à 0, pC possède d'autres propriétés élémentaires. Si C est convexe — cas le plus souvent étudié — pC est même sous-linéaire, mais elle n'est pas nécessairement symétrique et elle peut prendre des valeurs infinies. Sous certaines hypothèses supplémentaires, pC est une semi-norme dont C est la boule unité. Cette notion intervient en analyse fonctionnelle (démonstration de la forme analytique du théorème de Hahn-Banach), en optimisation (problème de recouvrement par jauge, optimisation conique), en apprentissage automatique, en géométrie des nombres (second théorème de Minkowski), etc. Dans tout cet article, E désigne un espace vectoriel réel, qu'on supposera topologique chaque fois que nécessaire. (fr)
- In mathematics, in the field of functional analysis, a Minkowski functional (after Hermann Minkowski) or gauge function is a function that recovers a notion of distance on a linear space. If is a subset of a real or complex vector space then the Minkowski functional or gauge of is defined to be the function valued in the extended real numbers, defined by where the infimum of the empty set is defined to be positive infinity (which is not a real number so that would then not be real-valued). The Minkowski function is always non-negative (meaning ) and is a real number if and only if is not empty. This property of being nonnegative stands in contrast to other classes of functions, such as sublinear functions and real linear functionals, that do allow negative values. In functional analysis, is usually assumed to have properties (such as being absorbing in for instance) that will guarantee that for every this set is not empty precisely because this results in being real-valued. Moreover, is also often assumed to have more properties, such as being an absorbing disk in since these properties guarantee that will be a (real-valued) seminorm on In fact, every seminorm on is equal to the Minkowski functional of any subset of satisfying (where all three of these sets are necessarily absorbing in and the first and last are also disks). Thus every seminorm (which is a function defined by purely algebraic properties) can be associated (non-uniquely) with an absorbing disk (which is a set with certain geometric properties) and conversely, every absorbing disk can be associated with its Minkowski functional (which will necessarily be a seminorm). These relationships between seminorms, Minkowski functionals, and absorbing disks is a major reason why Minkowski functionals are studied and used in functional analysis. In particular, through these relationships, Minkowski functionals allow one to "translate" certain geometric properties of a subset of into certain algebraic properties of a function on (en)
- 数学の関数解析学の分野におけるミンコフスキー汎関数（ミンコフスキーはんかんすう、英: Minkowski functional）とは、線型空間上に距離の概念をもたらすような関数のことである。 K を、線型空間 V に含まれる対称な凸体とする。V 上の関数 p を によって定める（ただしこの右辺が well-defined である場合）。 (ja)
- In matematica, in particolare in analisi funzionale, un funzionale di Minkowski è una funzione che richiama il concetto di distanza tipico degli spazi vettoriali. (it)
- Em matemática, sobretudo na análise funcional, um funcional de Minkowski faz uma interpretação geométrica dos funcionais norma e semi-norma. (pt)
- Funkcjonał Minkowskiego – i dodatnio jednorodny funkcjonał związany z i wypukłymi podzbiorami przestrzeni liniowej. (pl)
- У функціональному аналізі функціонал Мінковського використовує лінійну структуру простору для введення топології на ньому. Названий на честь німецького математика Германа Мінковського. (uk)
- Функционал Минковского — функционал, использующий линейную структуру пространства для введения топологии на нём. Назван по имени немецкого математика Германа Минковского. (ru)
- 度規函數是數學的一個重要函數。設為或上的向量空間，有需要時可以假設為拓撲向量空間。設為在內的凸集，且包含原點。那麼的度規函數是從到的函數，定義為 , 如果為空集，定義。 從定義立刻得到以下結果，可以進一步說明度規函數：
* 若是在中的開集，那麼；
* 若是在中的閉集，那麼。 (zh)
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- Assume that is an absorbing subset of
It is shown that:
# If is convex then is subadditive.
# If is balanced then is absolutely homogeneous; that is, for all scalars (en)
- If is a convex, balanced, and absorbing subset of a real or complex vector space then is a seminorm on (en)
- Let be any function.
The following statements are equivalent:
Strict positive homogeneity: for all and all real
* This statement is equivalent to: for all and all positive real
is a Minkowski functional; that is, there exists a subset such that
where
where
Moreover, if never takes on the value then this list may be extended to include:
Positive homogeneity/Nonnegative homogeneity: for all and all real (en)
- Let be any function and be any subset.
The following statements are equivalent:
is positive homogeneous, and
is the Minkowski functional of , contains the origin, and is star-shaped at the origin.
* The set is star-shaped at the origin if and only if whenever and A set that is star-shaped at the origin is sometimes called a . (en)
- Suppose that is a topological vector space over the real or complex numbers.
Then the non-empty open convex subsets of are exactly those sets that are of the form for some and some positive continuous sublinear function on (en)
- Let be a subset of a real or complex vector space
Then is a seminorm on if and only if all of the following conditions hold:
;
is convex;
* It suffices for to be convex.
for all unit scalars
* This condition is satisfied if is balanced or more generally if for all unit scalars
in which case and both and will be convex, balanced, and absorbing subsets of
Conversely, if is a seminorm on then the set satisfies all three of the above conditions and also
moreover, is necessarily convex, balanced, absorbing, and satisfies (en)
- If is an absorbing disk in a vector space then the Minkowski functional of which is the map defined by
is a seminorm on
Moreover, (en)
- Suppose that is a subset of a real or complex vector space
Strict positive homogeneity: for all and all real
* Positive/Nonnegative homogeneity: is nonnegative homogeneous if and only if is real-valued.
Real-values: is the set of all points on which is real valued. So is real-valued if and only if in which case
* Value at : if and only if if and only if
* Null space: If then if and only if if and only if there exists a divergent sequence of positive real numbers such that for all Moreover,
Comparison to a constant: If then for any this can be restated as: If then
* Thus if then where the set on the right hand side denotes and not its subset If then these sets are equal if and only if contains
* In particular, if then but importantly, the converse is not necessarily true.
Gauge comparison: For any subset thus if and only if
* The set satisfies so replacing with will not change the resulting Minkowski functional. The same is true of and of
* If then and has the particularly nice property that if is real then if and only if or Moreover, if is real then if and only if
Subadditive/Triangle inequality: is subadditive if and only if is convex. If is convex then so are both and and moreover, is subadditive.
Scaling the set: If is a scalar then for all
Thus if is real then
Absolute homogeneity: for all and all unit length scalars if and only if for all unit length scalars in which case for all and all scalars If in addition is also real-valued then this holds for scalars .
* for all unit length if and only if for all unit length
* for all unit scalars if and only if for all unit scalars if this is the case then for all unit scalars
* is symmetric if and only if which happens if and only if
* The Minkowski functional of any balanced set is a balanced function.
Absorbing: If is convex balanced and if then is absorbing in
* If a set is absorbing in and then is absorbing in
* If is convex and then in which case
Restriction to a vector subspace: If is a vector subspace of and if denotes the Minkowski functional of on then where denotes the restriction of to (en)
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- Summary (en)
- Theorem (en)
- Corollary (en)
- Proposition (en)
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- Proof (en)
- Proof that the Gauge of an absorbing disk is a seminorm (en)
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- Im mathematischen Teilgebiet der Funktionalanalysis ist das Minkowski-Funktional (nach Hermann Minkowski), oft auch Eichfunktional genannt, eine Verallgemeinerung des Normbegriffes. (de)
- 数学の関数解析学の分野におけるミンコフスキー汎関数（ミンコフスキーはんかんすう、英: Minkowski functional）とは、線型空間上に距離の概念をもたらすような関数のことである。 K を、線型空間 V に含まれる対称な凸体とする。V 上の関数 p を によって定める（ただしこの右辺が well-defined である場合）。 (ja)
- In matematica, in particolare in analisi funzionale, un funzionale di Minkowski è una funzione che richiama il concetto di distanza tipico degli spazi vettoriali. (it)
- Em matemática, sobretudo na análise funcional, um funcional de Minkowski faz uma interpretação geométrica dos funcionais norma e semi-norma. (pt)
- Funkcjonał Minkowskiego – i dodatnio jednorodny funkcjonał związany z i wypukłymi podzbiorami przestrzeni liniowej. (pl)
- У функціональному аналізі функціонал Мінковського використовує лінійну структуру простору для введення топології на ньому. Названий на честь німецького математика Германа Мінковського. (uk)
- Функционал Минковского — функционал, использующий линейную структуру пространства для введения топологии на нём. Назван по имени немецкого математика Германа Минковского. (ru)
- 度規函數是數學的一個重要函數。設為或上的向量空間，有需要時可以假設為拓撲向量空間。設為在內的凸集，且包含原點。那麼的度規函數是從到的函數，定義為 , 如果為空集，定義。 從定義立刻得到以下結果，可以進一步說明度規函數：
* 若是在中的開集，那麼；
* 若是在中的閉集，那麼。 (zh)
- In mathematics, in the field of functional analysis, a Minkowski functional (after Hermann Minkowski) or gauge function is a function that recovers a notion of distance on a linear space. If is a subset of a real or complex vector space then the Minkowski functional or gauge of is defined to be the function valued in the extended real numbers, defined by In functional analysis, is usually assumed to have properties (such as being absorbing in for instance) that will guarantee that for every this set is not empty precisely because this results in being real-valued. (en)
- En géométrie, la notion de jauge généralise celle de semi-norme. À toute partie C d'un ℝ-espace vectoriel E on associe sa jauge, ou fonctionnelle de Minkowski pC, qui est une application de E dans [0, +∞] mesurant, pour chaque vecteur, par quel rapport il faut dilater C pour englober ce vecteur. Dès que C contient l'origine, pC est positivement homogène ; si C est étoilée par rapport à 0, pC possède d'autres propriétés élémentaires. Si C est convexe — cas le plus souvent étudié — pC est même sous-linéaire, mais elle n'est pas nécessairement symétrique et elle peut prendre des valeurs infinies. Sous certaines hypothèses supplémentaires, pC est une semi-norme dont C est la boule unité. (fr)
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- Minkowski-Funktional (de)
- Fonctionnelle de Minkowski (fr)
- Funzionale di Minkowski (it)
- ミンコフスキー汎関数 (ja)
- Minkowski functional (en)
- Funkcjonał Minkowskiego (pl)
- Функционал Минковского (ru)
- Funcional de Minkowski (pt)
- Функціонал Мінковського (uk)
- 度規函數 (zh)
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