{"history":[],"article":{"id":"3be6b34f-b03e-4dd3-80b1-4e19faeb580f","title":"Absolutely convex set","slug":"absolutely-convex-set","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), a [subset](https://bliptext.com/articles/subset) C of a [real](https://bliptext.com/articles/real-number) or [complex](https://bliptext.com/articles/complex-number) [vector space](https://bliptext.com/articles/vector-space) is said to be **absolutely [convex](https://bliptext.com/articles/convex-set)** or ****disk**ed** if it is [convex](https://bliptext.com/articles/convex-set) and [balanced](https://bliptext.com/articles/balanced-set) (some people use the term \"circled\" instead of \"[balanced](https://bliptext.com/articles/balanced-set)\"), in which case it is called a **disk**.\nThe **disked hull** or the **absolute convex hull** of a set is the [intersection](https://bliptext.com/articles/intersection-set-theory) of all disks containing that set.\n\n# Definition\nA subset S of a real or complex vector space X is called a **' and is said to be '**, **', and '** if any of the following equivalent conditions is satisfied:\n

S is a [convex](https://bliptext.com/articles/convex-set) and [balanced set](https://bliptext.com/articles/balanced-set).
for any scalars a and b, if then
for all scalars a, b, and c, if then
for any scalars and c, if then
for any scalars if then

\nThe smallest [convex](https://bliptext.com/articles/convex-set) (respectively, [balanced](https://bliptext.com/articles/balanced-set)) subset of X containing a given set is called the [convex](https://bliptext.com/articles/convex-set) hull (respectively, the [balanced](https://bliptext.com/articles/balanced-set) hull) of that set and is denoted by (respectively, ).\nSimilarly, the **', the '**, and the of a set S is defined to be the smallest disk (with respect to subset [inclusion](https://bliptext.com/articles/inclusion-mathematics)) containing S. The disked hull of S will be denoted by or and it is equal to each of the following sets:\n

which is the convex hull of the [balanced hull](https://bliptext.com/articles/balanced-set) of S; thus,\n
the intersection of all disks containing S.

\n# [The light gray area is the absolutely convex hull of the cross. | upload.wikimedia.org/wikipedia/commons/0/0a/Absolute///convex///hull.svg]\n\n# Sufficient conditions\nThe intersection of arbitrarily many absolutely convex sets is again absolutely convex; however, [unions](https://bliptext.com/articles/union-set-theory) of absolutely convex sets need not be absolutely convex anymore.\nIf D is a disk in X, then D is absorbing in X if and only if\n\n# Properties\nIf S is an [absorbing](https://bliptext.com/articles/absorbing-set) disk in a vector space X then there exists an [absorbing](https://bliptext.com/articles/absorbing-set) disk E in X such that\nIf D is a disk and r and s are scalars then s D = |s| D and\nThe absolutely convex hull of a [bounded set](https://bliptext.com/articles/bounded-set-topological-vector-space) in a [locally convex topological vector space](https://bliptext.com/articles/locally-convex-topological-vector-space) is again bounded.\nIf D is a bounded disk in a TVS X and if is a [sequence](https://bliptext.com/articles/sequence) in D, then the partial sums are [Cauchy](https://bliptext.com/articles/cauchy-sequence), where for all n, In particular, if in addition D is a [sequentially complete](https://bliptext.com/articles/sequentially-complete) subset of X, then this series s_{\\bull} converges in X to some point of D.\nThe convex balanced hull of S contains both the convex hull of S and the balanced hull of S. Furthermore, it contains the balanced hull of the convex hull of S; thus\nwhere the example below shows that this inclusion might be strict.\nHowever, for any subsets if then which implies\n\n## Examples\nAlthough the convex balanced hull of S is necessarily equal to the balanced hull of the convex hull of S. For an example where let X be the real vector space \\R^2 and let\nThen is a strict subset of that is not even convex;\nin particular, this example also shows that the balanced hull of a convex set is necessarily convex.\nThe set is equal to the closed and filled square in X with vertices and (1, -1) (this is because the balanced set must contain both S and where since is also convex, it must consequently contain the solid square which for this particular example happens to also be balanced so that ).\nHowever, is equal to the horizontal closed line segment between the two points in S so that is instead a closed \"[hour glass](https://bliptext.com/articles/hour-glass) shaped\" subset that intersects the x-axis at exactly the origin and is the union of two closed and filled [isosceles triangles](https://bliptext.com/articles/isosceles-triangle): one whose vertices are the origin together with S and the other triangle whose vertices are the origin together with This non-convex filled \"hour-glass\" is a proper subset of the filled square\n\n# Generalizations\nGiven a fixed real number a is any subset C of a vector space X with the property that whenever c, d \\in C and r, s \\geq 0 are non-negative scalars satisfying\nIt is called an or a if whenever c, d \\in C and r, s are scalars satisfying\nA is any non-negative function that satisfies the following conditions:\nThis generalizes the definition of [seminorms](https://bliptext.com/articles/seminorm) since a map is a seminorm if and only if it is a 1-seminorm (using p := 1).\nThere exist p-[seminorms](https://bliptext.com/articles/seminorm) that are not [seminorms](https://bliptext.com/articles/seminorm).\nFor example, whenever 0 < p < 1 then the map used to define the [Lp space](https://bliptext.com/articles/lp-space) L_p(\\R) is a p-seminorm but not a seminorm.\nGiven a [topological vector space](https://bliptext.com/articles/topological-vector-space) is (meaning that its topology is induced by some p-seminorm) if and only if it has a [bounded](https://bliptext.com/articles/bounded-set-topological-vector-space) p-convex neighborhood of the origin.","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T21:35:56.419Z","updated_at":"2025-01-25T21:35:56.419Z","search_vector":"'absolut':1 'convex':2 'set':3","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), a [subset](https://bliptext.com/articles/subset) C of a [real](https://bliptext.com/articles/real-number) or [complex](https://bliptext.com/articles/complex-number) [vector space](https://bliptext.com/articles/vector-space) is said to be **absolutely [convex](https://bliptext.com/articles/convex-set)** or ****disk**ed** if it is [convex](https://bliptext.com/articles/convex-set) and [balanced](https://bliptext.com/articles/balanced-set) (some people use the term \"circled\" instead of \"[balanced](https://bliptext.com/articles/balanced-set)\"), in which case it is called a **disk**.\nThe **disked hull** or the **absolute convex hull** of a set is the [intersection](https://bliptext.com/articles/intersection-set-theory) of all disks containing that set.\n\n# Definition\nA subset S of a real or complex vector space X is called a **' and is said to be '**, **', and '** if any of the following equivalent conditions is satisfied:\n \n S is a [convex](https://bliptext.com/articles/convex-set) and [balanced set](https://bliptext.com/articles/balanced-set). \n for any scalars a and b, if then \n for all scalars a, b, and c, if then \n for any scalars and c, if then \n for any scalars if then \n \nThe smallest [convex](https://bliptext.com/articles/convex-set) (respectively, [balanced](https://bliptext.com/articles/balanced-set)) subset of X containing a given set is called the [convex](https://bliptext.com/articles/convex-set) hull (respectively, the [balanced](https://bliptext.com/articles/balanced-set) hull) of that set and is denoted by (respectively, ).\nSimilarly, the **', the '**, and the of a set S is defined to be the smallest disk (with respect to subset [inclusion](https://bliptext.com/articles/inclusion-mathematics)) containing S. The disked hull of S will be denoted by or and it is equal to each of the following sets:\n \n which is the convex hull of the [balanced hull](https://bliptext.com/articles/balanced-set) of S; thus,\n the intersection of all disks containing S. \n \n \n# [The light gray area is the absolutely convex hull of the cross. | upload.wikimedia.org/wikipedia/commons/0/0a/Absolute///convex///hull.svg]\n\n# Sufficient conditions\nThe intersection of arbitrarily many absolutely convex sets is again absolutely convex; however, [unions](https://bliptext.com/articles/union-set-theory) of absolutely convex sets need not be absolutely convex anymore.\nIf D is a disk in X, then D is absorbing in X if and only if\n\n# Properties\nIf S is an [absorbing](https://bliptext.com/articles/absorbing-set) disk in a vector space X then there exists an [absorbing](https://bliptext.com/articles/absorbing-set) disk E in X such that\nIf D is a disk and r and s are scalars then s D = |s| D and\nThe absolutely convex hull of a [bounded set](https://bliptext.com/articles/bounded-set-topological-vector-space) in a [locally convex topological vector space](https://bliptext.com/articles/locally-convex-topological-vector-space) is again bounded.\nIf D is a bounded disk in a TVS X and if is a [sequence](https://bliptext.com/articles/sequence) in D, then the partial sums are [Cauchy](https://bliptext.com/articles/cauchy-sequence), where for all n, In particular, if in addition D is a [sequentially complete](https://bliptext.com/articles/sequentially-complete) subset of X, then this series s_{\\bull} converges in X to some point of D.\nThe convex balanced hull of S contains both the convex hull of S and the balanced hull of S. Furthermore, it contains the balanced hull of the convex hull of S; thus\nwhere the example below shows that this inclusion might be strict.\nHowever, for any subsets if then which implies\n\n## Examples\nAlthough the convex balanced hull of S is necessarily equal to the balanced hull of the convex hull of S. For an example where let X be the real vector space \\R^2 and let\nThen is a strict subset of that is not even convex;\nin particular, this example also shows that the balanced hull of a convex set is necessarily convex.\nThe set is equal to the closed and filled square in X with vertices and (1, -1) (this is because the balanced set must contain both S and where since is also convex, it must consequently contain the solid square which for this particular example happens to also be balanced so that ).\nHowever, is equal to the horizontal closed line segment between the two points in S so that is instead a closed \"[hour glass](https://bliptext.com/articles/hour-glass) shaped\" subset that intersects the x-axis at exactly the origin and is the union of two closed and filled [isosceles triangles](https://bliptext.com/articles/isosceles-triangle): one whose vertices are the origin together with S and the other triangle whose vertices are the origin together with This non-convex filled \"hour-glass\" is a proper subset of the filled square\n\n# Generalizations\nGiven a fixed real number a is any subset C of a vector space X with the property that whenever c, d \\in C and r, s \\geq 0 are non-negative scalars satisfying\nIt is called an or a if whenever c, d \\in C and r, s are scalars satisfying\nA is any non-negative function that satisfies the following conditions:\nThis generalizes the definition of [seminorms](https://bliptext.com/articles/seminorm) since a map is a seminorm if and only if it is a 1-seminorm (using p := 1).\nThere exist p-[seminorms](https://bliptext.com/articles/seminorm) that are not [seminorms](https://bliptext.com/articles/seminorm).\nFor example, whenever 0 < p < 1 then the map used to define the [Lp space](https://bliptext.com/articles/lp-space) L_p(\\R) is a p-seminorm but not a seminorm.\nGiven a [topological vector space](https://bliptext.com/articles/topological-vector-space) is (meaning that its topology is induced by some p-seminorm) if and only if it has a [bounded](https://bliptext.com/articles/bounded-set-topological-vector-space) p-convex neighborhood of the origin.","revisionCount":0},"hasMore":false,"nextCursor":null}

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