. "Funkcjona\u0142 Minkowskiego \u2013 i dodatnio jednorodny funkcjona\u0142 zwi\u0105zany z i wypuk\u0142ymi podzbiorami przestrzeni liniowej."@pl . . . . . "Minkowski-Funktional"@de . . . . . . "Proposition"@en . . . . "41758"^^ . "Let be any function. \nThe following statements are equivalent: \n\nStrict positive homogeneity: for all and all real \n* This statement is equivalent to: for all and all positive real \n is a Minkowski functional; that is, there exists a subset such that \n where \n where \n\nMoreover, if never takes on the value then this list may be extended to include: \n\nPositive homogeneity/Nonnegative homogeneity: for all and all real"@en . "Proof"@en . . . . "Funkcjona\u0142 Minkowskiego \u2013 i dodatnio jednorodny funkcjona\u0142 zwi\u0105zany z i wypuk\u0142ymi podzbiorami przestrzeni liniowej."@pl . . "Im mathematischen Teilgebiet der Funktionalanalysis ist das Minkowski-Funktional (nach Hermann Minkowski), oft auch Eichfunktional genannt, eine Verallgemeinerung des Normbegriffes."@de . . . . . . "Em matem\u00E1tica, sobretudo na an\u00E1lise funcional, um funcional de Minkowski faz uma interpreta\u00E7\u00E3o geom\u00E9trica dos funcionais norma e semi-norma."@pt . . "\u0423 \u0444\u0443\u043D\u043A\u0446\u0456\u043E\u043D\u0430\u043B\u044C\u043D\u043E\u043C\u0443 \u0430\u043D\u0430\u043B\u0456\u0437\u0456 \u0444\u0443\u043D\u043A\u0446\u0456\u043E\u043D\u0430\u043B \u041C\u0456\u043D\u043A\u043E\u0432\u0441\u044C\u043A\u043E\u0433\u043E \u0432\u0438\u043A\u043E\u0440\u0438\u0441\u0442\u043E\u0432\u0443\u0454 \u043B\u0456\u043D\u0456\u0439\u043D\u0443 \u0441\u0442\u0440\u0443\u043A\u0442\u0443\u0440\u0443 \u043F\u0440\u043E\u0441\u0442\u043E\u0440\u0443 \u0434\u043B\u044F \u0432\u0432\u0435\u0434\u0435\u043D\u043D\u044F \u0442\u043E\u043F\u043E\u043B\u043E\u0433\u0456\u0457 \u043D\u0430 \u043D\u044C\u043E\u043C\u0443. \u041D\u0430\u0437\u0432\u0430\u043D\u0438\u0439 \u043D\u0430 \u0447\u0435\u0441\u0442\u044C \u043D\u0456\u043C\u0435\u0446\u044C\u043A\u043E\u0433\u043E \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u043A\u0430 \u0413\u0435\u0440\u043C\u0430\u043D\u0430 \u041C\u0456\u043D\u043A\u043E\u0432\u0441\u044C\u043A\u043E\u0433\u043E."@uk . . . . . . "\u5EA6\u898F\u51FD\u6578"@zh . "\u0424\u0443\u043D\u043A\u0446\u0438\u043E\u043D\u0430\u043B \u041C\u0438\u043D\u043A\u043E\u0432\u0441\u043A\u043E\u0433\u043E \u2014 \u0444\u0443\u043D\u043A\u0446\u0438\u043E\u043D\u0430\u043B, \u0438\u0441\u043F\u043E\u043B\u044C\u0437\u0443\u044E\u0449\u0438\u0439 \u043B\u0438\u043D\u0435\u0439\u043D\u0443\u044E \u0441\u0442\u0440\u0443\u043A\u0442\u0443\u0440\u0443 \u043F\u0440\u043E\u0441\u0442\u0440\u0430\u043D\u0441\u0442\u0432\u0430 \u0434\u043B\u044F \u0432\u0432\u0435\u0434\u0435\u043D\u0438\u044F \u0442\u043E\u043F\u043E\u043B\u043E\u0433\u0438\u0438 \u043D\u0430 \u043D\u0451\u043C. \u041D\u0430\u0437\u0432\u0430\u043D \u043F\u043E \u0438\u043C\u0435\u043D\u0438 \u043D\u0435\u043C\u0435\u0446\u043A\u043E\u0433\u043E \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u043A\u0430 \u0413\u0435\u0440\u043C\u0430\u043D\u0430 \u041C\u0438\u043D\u043A\u043E\u0432\u0441\u043A\u043E\u0433\u043E."@ru . "Funzionale di Minkowski"@it . "\u0423 \u0444\u0443\u043D\u043A\u0446\u0456\u043E\u043D\u0430\u043B\u044C\u043D\u043E\u043C\u0443 \u0430\u043D\u0430\u043B\u0456\u0437\u0456 \u0444\u0443\u043D\u043A\u0446\u0456\u043E\u043D\u0430\u043B \u041C\u0456\u043D\u043A\u043E\u0432\u0441\u044C\u043A\u043E\u0433\u043E \u0432\u0438\u043A\u043E\u0440\u0438\u0441\u0442\u043E\u0432\u0443\u0454 \u043B\u0456\u043D\u0456\u0439\u043D\u0443 \u0441\u0442\u0440\u0443\u043A\u0442\u0443\u0440\u0443 \u043F\u0440\u043E\u0441\u0442\u043E\u0440\u0443 \u0434\u043B\u044F \u0432\u0432\u0435\u0434\u0435\u043D\u043D\u044F \u0442\u043E\u043F\u043E\u043B\u043E\u0433\u0456\u0457 \u043D\u0430 \u043D\u044C\u043E\u043C\u0443. \u041D\u0430\u0437\u0432\u0430\u043D\u0438\u0439 \u043D\u0430 \u0447\u0435\u0441\u0442\u044C \u043D\u0456\u043C\u0435\u0446\u044C\u043A\u043E\u0433\u043E \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u043A\u0430 \u0413\u0435\u0440\u043C\u0430\u043D\u0430 \u041C\u0456\u043D\u043A\u043E\u0432\u0441\u044C\u043A\u043E\u0433\u043E."@uk . . . "Suppose that is a topological vector space over the real or complex numbers. \nThen 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 . "3510908"^^ . . . . . "If is an absorbing disk in a vector space then the Minkowski functional of which is the map defined by\n\nis a seminorm on \nMoreover,"@en . "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 g\u00E9om\u00E9trie, la notion de jauge g\u00E9n\u00E9ralise celle de semi-norme. \u00C0 toute partie C d'un \u211D-espace vectoriel E on associe sa jauge, ou fonctionnelle de Minkowski pC, qui est une application de E dans [0, +\u221E] mesurant, pour chaque vecteur, par quel rapport il faut dilater C pour englober ce vecteur. D\u00E8s que C contient l'origine, pC est positivement homog\u00E8ne ; si C est \u00E9toil\u00E9e par rapport \u00E0 0, pC poss\u00E8de d'autres propri\u00E9t\u00E9s \u00E9l\u00E9mentaires. Si C est convexe \u2014 cas le plus souvent \u00E9tudi\u00E9 \u2014 pC est m\u00EAme sous-lin\u00E9aire, mais elle n'est pas n\u00E9cessairement sym\u00E9trique et elle peut prendre des valeurs infinies. Sous certaines hypoth\u00E8ses suppl\u00E9mentaires, pC est une semi-norme dont C est la boule unit\u00E9."@fr . "\u0424\u0443\u043D\u043A\u0446\u0438\u043E\u043D\u0430\u043B \u041C\u0438\u043D\u043A\u043E\u0432\u0441\u043A\u043E\u0433\u043E \u2014 \u0444\u0443\u043D\u043A\u0446\u0438\u043E\u043D\u0430\u043B, \u0438\u0441\u043F\u043E\u043B\u044C\u0437\u0443\u044E\u0449\u0438\u0439 \u043B\u0438\u043D\u0435\u0439\u043D\u0443\u044E \u0441\u0442\u0440\u0443\u043A\u0442\u0443\u0440\u0443 \u043F\u0440\u043E\u0441\u0442\u0440\u0430\u043D\u0441\u0442\u0432\u0430 \u0434\u043B\u044F \u0432\u0432\u0435\u0434\u0435\u043D\u0438\u044F \u0442\u043E\u043F\u043E\u043B\u043E\u0433\u0438\u0438 \u043D\u0430 \u043D\u0451\u043C. \u041D\u0430\u0437\u0432\u0430\u043D \u043F\u043E \u0438\u043C\u0435\u043D\u0438 \u043D\u0435\u043C\u0435\u0446\u043A\u043E\u0433\u043E \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u043A\u0430 \u0413\u0435\u0440\u043C\u0430\u043D\u0430 \u041C\u0438\u043D\u043A\u043E\u0432\u0441\u043A\u043E\u0433\u043E."@ru . . . . "Em matem\u00E1tica, sobretudo na an\u00E1lise funcional, um funcional de Minkowski faz uma interpreta\u00E7\u00E3o geom\u00E9trica dos funcionais norma e semi-norma."@pt . . . "Funkcjona\u0142 Minkowskiego"@pl . . "\u30DF\u30F3\u30B3\u30D5\u30B9\u30AD\u30FC\u6C4E\u95A2\u6570"@ja . . . . . "Im mathematischen Teilgebiet der Funktionalanalysis ist das Minkowski-Funktional (nach Hermann Minkowski), oft auch Eichfunktional genannt, eine Verallgemeinerung des Normbegriffes."@de . . "\u5EA6\u898F\u51FD\u6578\u662F\u6578\u5B78\u7684\u4E00\u500B\u91CD\u8981\u51FD\u6578\u3002\u8A2D\u70BA\u6216\u4E0A\u7684\u5411\u91CF\u7A7A\u9593\uFF0C\u6709\u9700\u8981\u6642\u53EF\u4EE5\u5047\u8A2D\u70BA\u62D3\u64B2\u5411\u91CF\u7A7A\u9593\u3002\u8A2D\u70BA\u5728\u5167\u7684\u51F8\u96C6\uFF0C\u4E14\u5305\u542B\u539F\u9EDE\u3002\u90A3\u9EBC\u7684\u5EA6\u898F\u51FD\u6578\u662F\u5F9E\u5230\u7684\u51FD\u6578\uFF0C\u5B9A\u7FA9\u70BA , \u5982\u679C\u70BA\u7A7A\u96C6\uFF0C\u5B9A\u7FA9\u3002 \u5F9E\u5B9A\u7FA9\u7ACB\u523B\u5F97\u5230\u4EE5\u4E0B\u7D50\u679C\uFF0C\u53EF\u4EE5\u9032\u4E00\u6B65\u8AAA\u660E\u5EA6\u898F\u51FD\u6578\uFF1A \n* \u82E5\u662F\u5728\u4E2D\u7684\u958B\u96C6\uFF0C\u90A3\u9EBC\uFF1B \n* \u82E5\u662F\u5728\u4E2D\u7684\u9589\u96C6\uFF0C\u90A3\u9EBC\u3002"@zh . "Corollary"@en . "Theorem"@en . . . "En g\u00E9om\u00E9trie, la notion de jauge g\u00E9n\u00E9ralise celle de semi-norme. \u00C0 toute partie C d'un \u211D-espace vectoriel E on associe sa jauge, ou fonctionnelle de Minkowski pC, qui est une application de E dans [0, +\u221E] mesurant, pour chaque vecteur, par quel rapport il faut dilater C pour englober ce vecteur. D\u00E8s que C contient l'origine, pC est positivement homog\u00E8ne ; si C est \u00E9toil\u00E9e par rapport \u00E0 0, pC poss\u00E8de d'autres propri\u00E9t\u00E9s \u00E9l\u00E9mentaires. Si C est convexe \u2014 cas le plus souvent \u00E9tudi\u00E9 \u2014 pC est m\u00EAme sous-lin\u00E9aire, mais elle n'est pas n\u00E9cessairement sym\u00E9trique et elle peut prendre des valeurs infinies. Sous certaines hypoth\u00E8ses suppl\u00E9mentaires, pC est une semi-norme dont C est la boule unit\u00E9. Cette notion intervient en analyse fonctionnelle (d\u00E9monstration de la forme analytique du th\u00E9or\u00E8me de Hahn-Banach), en optimisation (probl\u00E8me de recouvrement par jauge, optimisation conique), en apprentissage automatique, en g\u00E9om\u00E9trie des nombres (second th\u00E9or\u00E8me de Minkowski), etc. Dans tout cet article, E d\u00E9signe un espace vectoriel r\u00E9el, qu'on supposera topologique chaque fois que n\u00E9cessaire."@fr . . . . "Let be a subset of a real or complex vector space \nThen is a seminorm on if and only if all of the following conditions hold:\n\n ;\n is convex;\n* It suffices for to be convex.\n for all unit scalars \n* This condition is satisfied if is balanced or more generally if for all unit scalars \n\nin which case and both and will be convex, balanced, and absorbing subsets of \n\nConversely, if is a seminorm on then the set satisfies all three of the above conditions and also \nmoreover, is necessarily convex, balanced, absorbing, and satisfies"@en . . . . . . . . . . . . . . . . . "\u0424\u0443\u043D\u043A\u0446\u0456\u043E\u043D\u0430\u043B \u041C\u0456\u043D\u043A\u043E\u0432\u0441\u044C\u043A\u043E\u0433\u043E"@uk . "If is a convex, balanced, and absorbing subset of a real or complex vector space then is a seminorm on"@en . . . . . "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 . "Funcional de Minkowski"@pt . "Assume that is an absorbing subset of \nIt is shown that: \n# If is convex then is subadditive.\n# If is balanced then is absolutely homogeneous; that is, for all scalars"@en . . . . "\u0424\u0443\u043D\u043A\u0446\u0438\u043E\u043D\u0430\u043B \u041C\u0438\u043D\u043A\u043E\u0432\u0441\u043A\u043E\u0433\u043E"@ru . "Proof that the Gauge of an absorbing disk is a seminorm"@en . . "\u6570\u5B66\u306E\u95A2\u6570\u89E3\u6790\u5B66\u306E\u5206\u91CE\u306B\u304A\u3051\u308B\u30DF\u30F3\u30B3\u30D5\u30B9\u30AD\u30FC\u6C4E\u95A2\u6570\uFF08\u30DF\u30F3\u30B3\u30D5\u30B9\u30AD\u30FC\u306F\u3093\u304B\u3093\u3059\u3046\u3001\u82F1: Minkowski functional\uFF09\u3068\u306F\u3001\u7DDA\u578B\u7A7A\u9593\u4E0A\u306B\u8DDD\u96E2\u306E\u6982\u5FF5\u3092\u3082\u305F\u3089\u3059\u3088\u3046\u306A\u95A2\u6570\u306E\u3053\u3068\u3067\u3042\u308B\u3002 K \u3092\u3001\u7DDA\u578B\u7A7A\u9593 V \u306B\u542B\u307E\u308C\u308B\u5BFE\u79F0\u306A\u51F8\u4F53\u3068\u3059\u308B\u3002V \u4E0A\u306E\u95A2\u6570 p \u3092 \u306B\u3088\u3063\u3066\u5B9A\u3081\u308B\uFF08\u305F\u3060\u3057\u3053\u306E\u53F3\u8FBA\u304C well-defined \u3067\u3042\u308B\u5834\u5408\uFF09\u3002"@ja . . . . . "Minkowski functional"@en . . "In matematica, in particolare in analisi funzionale, un funzionale di Minkowski \u00E8 una funzione che richiama il concetto di distanza tipico degli spazi vettoriali."@it . . "Fonctionnelle de Minkowski"@fr . . . . . . . . "Let be any function and be any subset. \nThe following statements are equivalent: \n\n is positive homogeneous, and \n\n is the Minkowski functional of , contains the origin, and is star-shaped at the origin.\n* 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 . "In matematica, in particolare in analisi funzionale, un funzionale di Minkowski \u00E8 una funzione che richiama il concetto di distanza tipico degli spazi vettoriali."@it . "\u5EA6\u898F\u51FD\u6578\u662F\u6578\u5B78\u7684\u4E00\u500B\u91CD\u8981\u51FD\u6578\u3002\u8A2D\u70BA\u6216\u4E0A\u7684\u5411\u91CF\u7A7A\u9593\uFF0C\u6709\u9700\u8981\u6642\u53EF\u4EE5\u5047\u8A2D\u70BA\u62D3\u64B2\u5411\u91CF\u7A7A\u9593\u3002\u8A2D\u70BA\u5728\u5167\u7684\u51F8\u96C6\uFF0C\u4E14\u5305\u542B\u539F\u9EDE\u3002\u90A3\u9EBC\u7684\u5EA6\u898F\u51FD\u6578\u662F\u5F9E\u5230\u7684\u51FD\u6578\uFF0C\u5B9A\u7FA9\u70BA , \u5982\u679C\u70BA\u7A7A\u96C6\uFF0C\u5B9A\u7FA9\u3002 \u5F9E\u5B9A\u7FA9\u7ACB\u523B\u5F97\u5230\u4EE5\u4E0B\u7D50\u679C\uFF0C\u53EF\u4EE5\u9032\u4E00\u6B65\u8AAA\u660E\u5EA6\u898F\u51FD\u6578\uFF1A \n* \u82E5\u662F\u5728\u4E2D\u7684\u958B\u96C6\uFF0C\u90A3\u9EBC\uFF1B \n* \u82E5\u662F\u5728\u4E2D\u7684\u9589\u96C6\uFF0C\u90A3\u9EBC\u3002"@zh . . . . . "\u6570\u5B66\u306E\u95A2\u6570\u89E3\u6790\u5B66\u306E\u5206\u91CE\u306B\u304A\u3051\u308B\u30DF\u30F3\u30B3\u30D5\u30B9\u30AD\u30FC\u6C4E\u95A2\u6570\uFF08\u30DF\u30F3\u30B3\u30D5\u30B9\u30AD\u30FC\u306F\u3093\u304B\u3093\u3059\u3046\u3001\u82F1: Minkowski functional\uFF09\u3068\u306F\u3001\u7DDA\u578B\u7A7A\u9593\u4E0A\u306B\u8DDD\u96E2\u306E\u6982\u5FF5\u3092\u3082\u305F\u3089\u3059\u3088\u3046\u306A\u95A2\u6570\u306E\u3053\u3068\u3067\u3042\u308B\u3002 K \u3092\u3001\u7DDA\u578B\u7A7A\u9593 V \u306B\u542B\u307E\u308C\u308B\u5BFE\u79F0\u306A\u51F8\u4F53\u3068\u3059\u308B\u3002V \u4E0A\u306E\u95A2\u6570 p \u3092 \u306B\u3088\u3063\u3066\u5B9A\u3081\u308B\uFF08\u305F\u3060\u3057\u3053\u306E\u53F3\u8FBA\u304C well-defined \u3067\u3042\u308B\u5834\u5408\uFF09\u3002"@ja . . "true"@en . "1108213796"^^ . . . . . . . "Summary"@en . . . "Suppose that is a subset of a real or complex vector space \n\nStrict positive homogeneity: for all and all real \n* Positive/Nonnegative homogeneity: is nonnegative homogeneous if and only if is real-valued.\nReal-values: is the set of all points on which is real valued. So is real-valued if and only if in which case \n* Value at : if and only if if and only if \n* 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, \n\nComparison to a constant: If then for any this can be restated as: If then \n* 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 \n* In particular, if then but importantly, the converse is not necessarily true. \nGauge comparison: For any subset thus if and only if \n* The set satisfies so replacing with will not change the resulting Minkowski functional. The same is true of and of \n* 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 \n\nSubadditive/Triangle inequality: is subadditive if and only if is convex. If is convex then so are both and and moreover, is subadditive.\nScaling the set: If is a scalar then for all \nThus if is real then \nAbsolute 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 .\n* for all unit length if and only if for all unit length \n* for all unit scalars if and only if for all unit scalars if this is the case then for all unit scalars \n* is symmetric if and only if which happens if and only if \n* The Minkowski functional of any balanced set is a balanced function. \n\nAbsorbing: If is convex balanced and if then is absorbing in \n* If a set is absorbing in and then is absorbing in \n* If is convex and then in which case \nRestriction 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 . . . . . . . .