Absolutely convex set
In mathematics, a subset "C" of a real or complex vector space is said to be absolutely convex or disked if it is convex and balanced (some people use the term "circled" instead of "balanced"), in which case it is called a disk. The disked hull or the absolute convex hull of a set is the intersection of all disks containing that set. Definition. A subset formula_1 of a real or complex vector space formula_2 is called a ' and is said to be ', ', and ' if any of the following equivalent conditions is satisfied: The smallest convex (resp . balanced) subset of formula_2 containing a set is called the convex hull (resp. balanced hull) of that set and is denoted by formula_4 (resp. formula_5). Similarly, the ', the ', and the "" of a set formula_1 is defined to be the smallest disk (with respect to subset inclusion) containing formula_7 The disked hull of formula_1 will be denoted by formula_9 or formula_10 and it is equal to each of the following sets: Sufficient conditions. The intersection of arbitrarily many absolutely convex sets is again absolutely convex; however, unions of absolutely convex sets need not be absolutely convex anymore. If formula_11 is a  disk in formula_12 then formula_11 is absorbing in formula_2 if and only if formula_15 Properties. If formula_1 is an absorbing disk in a vector space formula_2 then there exists an absorbing disk formula_18 in formula_2 such that formula_20 The absolutely convex hull of a bounded set in a locally convex topological vector space is again bounded. If formula_11 is a bounded disk in a TVS formula_2 and if formula_23 is a sequence in formula_24 then the partial sums formula_25 are Cauchy, where for all formula_26 formula_27 In particular, if in addition formula_11 is a sequentially complete subset of formula_12 then  this series formula_30 converges in formula_2 to some point of formula_32 The convex balanced hull of formula_1 contains both the convex hull of formula_1 and the balanced hull of formula_7 Furthermore, it contains the balanced hull of the convex hull of formula_36 thus formula_37 where the example below shows that this inclusion might be strict. However, for any subsets formula_38 if formula_39 then formula_40 which implies formula_41 Examples. Although formula_42 the convex balanced hull of formula_1 is necessarily equal to the balanced hull of the convex hull of formula_7 For an example where formula_45 let formula_2 be the real vector  space formula_47 and let formula_48 Then formula_49 is a strict subset of formula_10 that is not even convex; in particular, this example also shows that the balanced hull of a convex set is necessarily convex. The set formula_10 is equal to the closed and filled square in formula_2 with vertices formula_53 and formula_54 (this is because the balanced set formula_10 must contain both formula_1 and formula_57 where since formula_10 is also convex, it must consequently contain the solid square formula_59 which for this particular example happens to also be balanced so that formula_60). However, formula_61 is equal to the horizontal  closed line segment between the two points in formula_1 so that formula_49 is instead a closed "hour glass shaped" subset that intersects the formula_64-axis at exactly the origin and is the union of two closed and filled isosceles triangles: one whose vertices are the origin together with formula_1 and the other triangle whose vertices are the origin together with formula_66 This non-convex filled "hour-glass" formula_49 is a proper subset of the filled square formula_68