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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.

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rdf:type	yago:WikicatTopologicalVectorSpaces yago:Abstraction100002137 yago:Attribute100024264 yago:Space100028651
rdfs:label	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)
rdfs:comment	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)
name	Summary (en) Theorem (en) Corollary (en) Proposition (en)
foaf:depiction
dcterms:subject	Convex analysis Functional analysis Hermann Minkowski
Wikipage page ID	3510908 (xsd:integer)
Wikipage revision ID	1108213796 (xsd:integer)
Link from a Wikipage to another Wikipage	Cambridge University Press Topological interior Homogeneous function Vector space Infimum and supremum Complex number Convex analysis Convex set Mathematics Norm (mathematics) Closure (topology) Function (mathematics) Linear form Linear functionals Locally convex topological vector space Strict positive homogeneity Subadditive Sublinear function Empty set Functional analysis Balanced set Topological vector space Hausdorff space Absolute homogeneity Absolutely convex set Algebraic interior Extended real number line Nonnegative homogeneity Hermann Minkowski Absorbing set Convex analysis Functional analysis Hermann Minkowski Positive homogeneity Real number Seminorm Infimum Locally convex Normed vector space Star domain Restriction of a function Null space Positive infinity Balanced function Subadditive function Sublinear functional
Link from a Wikipage to an external page	https://archive.org/details/minkowskigeometr0000thom%7Curl-access=registration%7Cseries=Encyclopedia https://doi.org/10.3390/en13225976
sameAs	Minkowski functional Minkowski functional Minkowski functional Minkowski functional Minkowski functional Minkowski functional Minkowski functional Minkowski functional Minkowski functional Minkowski functional Minkowski functional Minkowski functional Minkowski functional
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