exists a positive real such that The proof that a balanced set satisfying this condition is necessarily absorbing in is almost immediate from the definition of a "balanced set". The proof that a convex set satisfying this condition is necessarily absorbing in is less trivial (but not difficult). A detailed proof is given in this footnote and a summary is given below. Summary of proof: By assumption, for non-zero it is possible to pick positive real and such that and so that the convex set contains the open sub-interval which contains the origin ( is called an interval since we identify with and every non-empty convex subset of is an interval). Give its unique Hausdorff vector topology so it remains to show that is a neighborhood of the origin in If then we are done, so assume that The set is a union of two intervals, each of which contains an open sub-interval that contains the origin; moreover, the intersection of these two intervals is precisely the origin. So the convex hull of which is contained in the convex set clearly contains an open ball around the origin. For every there exists a positive real such that This condition is equivalent to: every belongs to the set This happens if and only if which gives the next characterization. It can be shown that for any subset of if and only if for every where For every If (which is necessary for to be absorbing) then it suffices to check any of the above conditions for all non-zero rather than all Examples and sufficient conditions For one set to absorb another Let be a linear map between vector spaces and let and be balanced sets. Then absorbs if and only if absorbs If a set absorbs another set then any superset of also absorbs A set absorbs the origin if and only if the origin is an element of A set absorbs a finite union of sets if and only it absorbs each set individuality (that is, if and only if absorbs for every ). In particular, a set is an absorbing subset of if and only if it absorbs every finite subset of For a set to be absorbing In a semi normed vector space the unit ball is absorbing. More generally, if is a topological vector space (TVS) then any neighborhood of the origin in is absorbing in This