Balanced set

Definition

1. is the smallest balanced subset of containing ;
2. is the intersection of all balanced sets containing S;
3. .

1. is the largest balanced set contained in ;
2. is the union of all balanced subsets of ;
3. if while if.

   Examples and sufficient conditions

• The closure of a balanced set is balanced.
• The convex hull of a balanced set is convex and balanced.
  
  • However, the balanced hull of a convex set may fail to be convex.
• The balanced hull of a compact set is compact.
• Arbitrary unions of balanced sets are a balanced set.
• Arbitrary intersections of balanced sets are a balanced set.
• Scalar multiples of balanced sets are balanced.
• The Minkowski sum of two balanced sets is balanced.
• The image of a balanced set under a linear operator is again a balanced set.
• The inverse image of a balanced set under a linear operator is again a balanced set.
• In any topological vector space, the interior of a balanced neighborhood of 0 is again balanced.

• The open and closed balls centered at 0 in a normed vector space are balanced sets.
  
• Any vector subspace of a real or complex vector space is a balanced set.
• The cartesian product of a family of balanced sets is balanced in the product space of the corresponding vector spaces.
• Consider, the field of complex numbers, as a 1-dimensional vector space. The balanced sets are itself, the empty set and the open and closed discs centered at zero. Contrariwise, in the two dimensional Euclidean space there are many more balanced sets: any line segment with midpoint at the origin will do. As a result, and are entirely different as far as scalar multiplication is concerned.
• If is a semi-norm on a linear space, then for any constant the set is balanced.
• Let and let be the union of the line segment between and and the line segment between and. Then is balanced but not convex or absorbing. However,.
• Let and for every, let be any positive real number and let be the line segment between the points and. Then the set is balanced and absorbing but it is not necessarily convex.
• The balanced hull of a closed set need not be closed. Take for instance the graph of in.

Properties

• A set is absolutely convex if and only if it is convex and balanced
• If is balanced then for any scalar,.
• If is balanced then for any scalars and such that,.
• The union of and the interior of a balanced set is balanced.
• If is a balanced subset of, then is absorbing if and only if for all, there exists such that.
• The Minkowski sum of two balanced sets is balanced.

• for any subset of and any scalar.
• for any collection of subsets of.
• In any topological vector space, the balanced hull of any open neighborhood of 0 is again open.
• If is a Hausdorff topological vector space and if is a compact subset of, then the balanced hull of is compact.

• The balanced core of a closed subset is closed.
• The balanced core of a absorbing subset is absorbing.