convex balanced hull of is necessarily equal to the balanced hull of the convex hull of For an example where let be the real vector space and let Then is a strict subset of that is not even convex; in particular, this example also shows that the balanced hull of a convex set is necessarily convex. The set is equal to the closed and filled square in with vertices and (this is because the balanced set must contain both 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 ). However, is equal to the horizontal closed line segment between the two points in so that is instead a closed "hour glass shaped" subset that intersects the -axis at exactly the origin and is the union of two closed and filled isosceles triangles: one whose vertices are the origin together with 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 Generalizations Given a fixed real number a is any subset of a vector space with the property that whenever and are non-negative scalars satisfying It is called an or a if whenever and are scalars satisfying A is any non-negative function that satisfies the following conditions: Subadditivity/Triangle inequality: for all Absolute homogeneity of degree : for all and all scalars This generalizes the definition of seminorms since a map is a seminorm if and only if it is a -seminorm (using ). There exist -seminorms that are not seminorms. For example, whenever then the map used to define the Lp space is a -seminorm but not a seminorm. Given a topological vector space is (meaning that its topology is induced by some -seminorm) if and only if it has a bounded -convex neighborhood of the origin. See also , for vectors in physics References Bibliography Abstract algebra Convex analysis Convex geometry Group theory Linear algebra