Введение
Функция, заданная на множестве
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 set is often assumed/picked to have properties, such as being an absorbing disk in that guarantee that will be a real valued seminorm on
In fact, every seminorm on is equal to the Minkowski functional (that is, ) 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
The Minkowski function is always non negative (meaning ). 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. However, might not be real valued since for any given the value is a real number if and only if is not empty. Consequently, is usually assumed to have properties (such as being absorbing in for instance) that will guarantee that is real valued.
В математике, в области функционального анализа, функционал Минковского (в честь Германа Минковского) или калибровочная функция – это функция, восстанавливающая понятие расстояния на линейном пространстве. Если является подмножеством вещественного или комплексного векторного пространства , то функционал Минковского или калибр множества определяется как функция, принимающая значения в расширенных вещественных числах, заданная выражением
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 set is often assumed/picked to have properties, such as being an absorbing disk in that guarantee that will be a real valued seminorm on
In fact, every seminorm on is equal to the Minkowski functional (that is, ) 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
The Minkowski function is always non negative (meaning ). 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. However, might not be real valued since for any given the value is a real number if and only if is not empty. Consequently, is usually assumed to have properties (such as being absorbing in for instance) that will guarantee that is real valued.
где инфимум пустого множества определяется как положительная бесконечность (которое не является вещественным числом, поэтому тогда не будет принимать вещественные значения). Часто предполагается, что множество обладает свойствами, такими как свойство быть поглощающим диском в , которые гарантируют, что будет вещественнозначным семинармом на .
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 set is often assumed/picked to have properties, such as being an absorbing disk in that guarantee that will be a real valued seminorm on
In fact, every seminorm on is equal to the Minkowski functional (that is, ) 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
The Minkowski function is always non negative (meaning ). 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. However, might not be real valued since for any given the value is a real number if and only if is not empty. Consequently, is usually assumed to have properties (such as being absorbing in for instance) that will guarantee that is real valued.
В действительности, любой семинарм на равен функционалу Минковского (то есть ) любого подмножества из удовлетворяющего (где все три этих множества обязательно поглощающие в , а первое и последнее также являются дисками). Таким образом, любой семинарм (который является функцией, определяемой чисто алгебраическими свойствами) может быть сопоставлен (не единственным образом) поглощающему диску (который является множеством с определенными геометрическими свойствами), и наоборот, любому поглощающему диску может быть сопоставлен его функционал Минковского (который обязательно будет семинармом). Эти связи между семинармами, функционалами Минковского и поглощающими дисками являются основной причиной, по которой функционалы Минковского изучаются и используются в функциональном анализе. В частности, благодаря этим связям, функционалы Минковского позволяют "переводить" определенные геометрические свойства подмножества в определенные алгебраические свойства функции на .
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 set is often assumed/picked to have properties, such as being an absorbing disk in that guarantee that will be a real valued seminorm on
In fact, every seminorm on is equal to the Minkowski functional (that is, ) 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
The Minkowski function is always non negative (meaning ). 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. However, might not be real valued since for any given the value is a real number if and only if is not empty. Consequently, is usually assumed to have properties (such as being absorbing in for instance) that will guarantee that is real valued.
Функционал Минковского всегда неотрицателен (то есть ). Это свойство неотрицательности контрастирует с другими классами функций, такими как сублинейные функции и вещественные линейные функционалы, которые допускают отрицательные значения. Однако, может не принимать вещественные значения, поскольку для любого заданного значение является вещественным числом тогда и только тогда, когда не пусто. Следовательно, обычно предполагается, что обладает свойствами (например, свойством быть поглощающим в ), которые гарантируют, что будет вещественнозначным.
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 set is often assumed/picked to have properties, such as being an absorbing disk in that guarantee that will be a real valued seminorm on
In fact, every seminorm on is equal to the Minkowski functional (that is, ) 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
The Minkowski function is always non negative (meaning ). 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. However, might not be real valued since for any given the value is a real number if and only if is not empty. Consequently, is usually assumed to have properties (such as being absorbing in for instance) that will guarantee that is real valued.
Характеристика функционалов Минковского, которые являются семинармами
В следующей теореме, которая непосредственно вытекает из вышеприведенных утверждений, не предполагается, что является поглощающим в , а вместо этого доказывается, что оно поглощающее, когда является семинормой. Также не предполагается, что оно сбалансировано (свойство, которое часто требуется); вместо этого используется более слабое условие, что для всех скаляров , удовлетворяющих . Общее требование, чтобы было выпуклым, также ослаблено до требования только выпуклости .
The common requirement that be convex is also weakened to only requiring that be convex.