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 of a real or complex vector space is called a and is said to be , , and if any of the following equivalent conditions is satisfied: is a convex and balanced set. for any scalar and if then for all scalars and if then for any scalars and if then for any scalars if then The smallest convex (respectively, balanced) subset of containing a given set is called the convex hull (respectively, the balanced hull) of that set and is denoted by (respectively, ). Similarly, the , the , and the of a set is defined to be the smallest disk (with respect to subset inclusion) containing The disked hull of will be denoted by or and it is equal to each of the following sets: which is the convex hull of the balanced hull of ; thus, In general, is possible, even in finite dimensional vector spaces. the intersection of all disks containing 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 is a disk in then is absorbing in if and only if Properties If is an absorbing disk in a vector space then there exists an absorbing disk in such that If is a disk and and are scalars then and The absolutely convex hull of a bounded set in a locally convex topological vector space is again bounded. If is a bounded disk in a TVS and if is a sequence in then the partial sums are Cauchy, where for all In particular, if in addition is a sequentially complete subset of then this series converges in to some point of The convex balanced hull of contains both the convex hull of and the balanced hull of Furthermore, it contains the balanced hull of the convex hull of thus where the example below shows that this inclusion might be strict. However, for any subsets if then which implies Examples Although the