The balanced hull or balanced envelope of a set is the smallest balanced set containing The balanced core of a subset is the largest balanced set contained in
Notation: If is a set, is a scalar, and then let
and for any let
denote the closed ball (respectively, the open ball) of radius in centered at where and Every balanced subset of the field is of the form or for some
A subset of is called balanced if it satisfies any of the following equivalent conditions:
- Definition: for all scalars satisfying ;
- where ;
- For every ;
- If then the above equality becomes which is exactly the previous condition for a set to be balanced. Thus, is balanced if and only if for every is a balanced set (according to any of the previous defining conditions);
- For every 1-dimensional vector subspace of is a balanced set (according to any defining condition other than this one).
- For every there exists some such that or ;
If is a convex set then this list may be extended to include:
- for all scalars satisfying 
If then this list may be extended to include:
- is symmetric (meaning ) and
The balanced hull of a subset of denoted by is defined in any of the following equivalent ways:
- Definition: is the smallest (with respect to ) balanced subset of containing ;
- is the intersection of all balanced sets containing ;
The balanced core of a subset of denoted by is defined in any of the following equivalent ways:
- Definition: is the largest (with respect to ) balanced subset of ;
- is the union of all balanced subsets of ;
- if while if
Examples and sufficient conditions
- Sufficient conditions
- The closure of a balanced set is balanced.
- The convex hull of a balanced set is convex and balanced (also known as an absolutely convex set).
- However, the balanced hull of a convex set may fail to be convex.
- The balanced hull of a compact (resp. totally bounded, bounded) set is compact (resp. totally bounded, bounded).
- 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 (in the codomain) under a linear operator is again a balanced set (in the domain).
- In any topological vector space, the interior of a balanced neighborhood of the origin is again balanced.
- If is any subset and then is a balanced set.
- In particular, if is any balanced neighborhood of the origin in a TVS then
- If is the field real or complex numbers and is the normed space over with the usual Euclidean norm, then the balanced subsets of are exactly the following:
- for some real
- for some real
- 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.
- If ( is a vector space over ), is the closed unit ball in centered at the origin, is non-zero, and then the set is a closed, symmetric, and balanced neighborhood of the origin in More generally, if is any closed subset of such that then is a closed, symmetric, and balanced neighborhood of the origin in This example can be generalized to for any integer
- The Cartesian product of a family of balanced sets is balanced in the product space of the corresponding vector spaces (over the same field ).
- 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 seminorm 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 (open or closed) 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 of balanced sets
- 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 in if and only if for all there exists such that 
- If is a balanced subset of then is absorbing in
- The Minkowski sum of two balanced sets is balanced.
- Every balanced set is symmetric.
- Every balanced set is path connected.
- Suppose is balanced. If is a 1-dimensional vector subspace of then } is convex and balanced. If is a 1-dimensional vector subspace of then } is also absorbing in
- If is a balanced then for any is a convex balanced set containing the origin. If is a neighborhood of in then is a convex balanced neighborhood of in the real vector subspace
- Properties of balanced hulls
- 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 the origin is again open.
- If is a Hausdorff topological vector space and if is a compact subset of then the balanced hull of is compact.
- Balanced core
- The balanced core of a closed subset is closed.
- The balanced core of a absorbing subset is absorbing.
- Absolutely convex set
- Absorbing set – A set that can be "inflated" to eventually always include any given point in a space
- Bounded set (topological vector space)
- Convex set – In geometry, set that intersects every line into a single line segment
- Star domain
- Symmetric set
- Topological vector space – Vector space with a notion of nearness
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