set will be convex (respectively, closed, balanced, bounded, a neighborhood of the origin, an absorbing subset of ) whenever this is true of It will be a balanced set if is a star shaped at the origin, which is true, for instance, when is convex and contains In particular, if is a convex neighborhood of the origin then will be a convex neighborhood of the origin and so its topological interior will be a balanced convex neighborhood of the origin. Suppose that is a convex and absorbing subset of Then will be convex balanced absorbing subset of which guarantees that the Minkowski functional of will be a seminorm on thereby making into a seminormed space that carries its canonical pseduometrizable topology. The set of scalar multiples as ranges over (or over any other set of non-zero scalars having as a limit point) forms a neighborhood basis of absorbing disks at the origin for this locally convex topology. If is a topological vector space and if this convex absorbing subset is also a bounded subset of then the same will be true of the absorbing disk in which case will be a norm and will form what is known as an auxiliary normed space. If this normed space is a Banach space then is called a . Properties Properties of balanced sets A balanced set is not empty if and only if it contains the origin. By definition, a set is absolutely convex if and only if it is convex and balanced. Every balanced set is star-shaped (at 0) and a symmetric set. If is a balanced subset of then If is a balanced subset of then: for any scalars and if then and Thus if and are any scalars then is absorbing in if and only if for all there exists such that for any 1-dimensional vector subspace of the set is convex and balanced. If is not empty and if is a 1-dimensional vector subspace of then is either or else it is absorbing in for any if contains more than one point then it is a convex and balanced neighborhood of in the 1-dimensional vector space when this space is endowed with the Hausdorff Euclidean topology; and the set is a convex balanced subset of the real vector space that contains the origin. Properties of balanced hulls and balanced cores For any collection of subsets of In any