Balanced set

Suppose that X is a vector space over the field 𝕂 of real or complex numbers. Elements of 𝕂 are called scalars.

Notation: If S is a set, a is a scalar, and B ⊆ 𝕂 then let a S := { a s : s ∈ S } and B S := { b s : b ∈ B, s ∈ S }.

Notation: For any 0 ≤ r ≤ ∞, let B_r := { a ∈ 𝕂 : |a| ≤ r } denote the closed ball of radius r in 𝕂 centered at 0 and let B_r := { a ∈ 𝕂 : |a| < r } denote the corresponding open ball. Note that B₀ = ∅, B₀ = { 0  }, and B_∞ = B_∞ = 𝕂.

• Observe that every balanced subset of the field 𝕂 is of the form B_r or B_r for some 0 ≤ r ≤ ∞.

Definition: A subset S of X is called balanced if it satisfies any of the following equivalent conditions:

1. a S ⊆ S for all scalars a satisfying |a| ≤ 1;
2. B₁ S ⊆ S, where B₁ := { a ∈ 𝕂 : |a| ≤ 1 };
3. S = B₁ S;[1]
4. For every s ∈ S, S ∩ 𝕂 s = B₁ (S ∩ 𝕂 s);
   • Note that if we let R := S ∩ 𝕂 s then the above equality becomes R = B₁ R, which is exactly the previous condition for a set to be balanced. Thus, S is balanced if and only if for every s ∈ S, S ∩ 𝕂 s is a balanced set (according to any of the previous defining conditions);
5. For every 1-dimensional vector subspace Y of span S, S ∩ Y is a balanced set (according to any defining condition over than this one).
6. For every s ∈ S, there exists some 0 ≤ r ≤ ∞ such that S ∩ 𝕂 s = B_r s or S ∩ 𝕂 s = B_r s;

If S is a convex set then we may add to this list:

7. a S ⊆ S for all scalars a satisfying |a| = 1.[2]

If 𝕂 = ℝ then we may add to this list:

8. S is symmetric (i.e. - S = S) and [0, 1) S ⊆ S.

Definition and notation: The balanced hull of a subset S of X, denoted by bal S, is defined in any of the following equivalent ways:

1. bal S is the smallest balanced subset of X containing S;
2. bal S is the intersection of all balanced sets containing S;
3. bal S = ∪|a| ≤ 1 (a S);
4. bal S = B₁ S, where B₁ := { a ∈ 𝕂 : |a| ≤ 1 }.[1]

Definition and notation: The balanced core of a subset S of X, denoted by balcore S, is defined in any of the following equivalent ways:

1. balcore S is the largest balanced set contained in S;
2. balcore S is the union of all balanced subsets of S;
3. balcore S = ∅ if 0 ∉ S while balcore S = ∩|a| ≥ 1 (aS) if 0 ∈ S.

Sufficient conditions

• The closure of a balanced set is balanced.
• The convex hull of a balanced set is convex and balanced (i.e. absolutely convex).
  • 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).[3]
• 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 0 is again balanced.

Examples

• If S ⊆ X is any subset and B₁ := { a ∈ 𝕂 : |a| < 1 } then B₁ S is a balanced set.
  • In particular, if U ⊆ X is any balanced neighborhood of the origin in a TVS X then Int U = B₁ U = ∪0 < |a| < 1 a U ⊆ U.
• If 𝕂 is the field real or complex numbers and X = 𝕂 is the normed space over 𝕂 with the usual Euclidean norm, then the balanced subsets of X are exactly the following:[4]
  1. ∅
  2. X
  3. { 0 }
  4. { x ∈ X : |x| < r } for some real r > 0
  5. { x ∈ X : |x| ≤ r } for some real r > 0.
• 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 X = ℝ² (X is a vector space over ℝ), B₁ is the closed unit ball in X centered at the origin, x₀ ∈ X is non-zero, and L := ℝx₀, then the set R := B₁ ∪ L is a closed, symmetric, and balanced neighborhood of the origin in X. More generally, if C is any closed subset of X such that (0, 1) C ⊆ C, then S := B₁ ∪ C ∪ (- C) is a closed, symmetric, and balanced neighborhood of the origin in X. This example can be generalized to ℝⁿ for any integer n ≥ 1.
• The cartesian product of a family of balanced sets is balanced in the product space of the corresponding vector spaces (over the same field 𝕂).
• Consider ℂ, the field of complex numbers, as a 1-dimensional vector space. The balanced sets are ℂ 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, ℂ and ℝ² are entirely different as far as scalar multiplication is concerned.
• If p is a semi-norm on a linear space X, then for any constant c > 0 the set { x ∈ X : p(x) ≤ c } is balanced.
• Let X = ℝ² and let B be the union of the line segment between (-1, 0) and (1, 0) and the line segment between (0, -1) and (0, 1). Then B is balanced but not convex or absorbing. However, span B = X.
• Let X = ℝ² and for every 0 ≤ t < 𝜋, let r_t be any positive real number and let B^t be the (open or closed) line segment between the points (cos t, sin t) and -(cos t, sin t). Then the set B = ∪0 ≤ t < 𝜋 r_t B^t 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 xy = 1 in X = ℝ².

Properties of balanced sets

• A set is absolutely convex if and only if it is convex and balanced.
• If B is balanced then for any scalar a, aB = |a|B.
• If B is balanced then for any scalars a and b such that |a| ≤ |b|, aB ⊆ bB.
• The union of { 0 } and the interior of a balanced set is balanced.
• If B is a balanced subset of X, then B is absorbing in X if and only if for all x ∈ X, there exists r > 0 such that x ∈ rB.[2]
• If B is a balanced subset of X, then B is absorbing in span B.
• The Minkowski sum of two balanced sets is balanced.
• Every balanced set is symmetric.
• Every balanced set is path connected.
• Suppose B is balanced. If Y is a 1-dimensional vector subspace of X then B ∩ Y is convex and balanced. If Y is a 1-dimensional vector subspace of span B then B ∩ Y is also absorbing in Y.
• If B ≠ ∅ is a balanced then for any x ∈ X, B ∩ ℝ x is a convex balanced set containing the origin. If B is a neighborhood of 0 in X then B ∩ ℝ x is a convex balanced neighborhood of 0 in the real vector subspace ℝ x.

Properties of balanced hulls

• a bal S = bal(aS) for any subset S of X and any scalar a.
• bal(∪S ∈ 𝒮 S) = ∪S ∈ 𝒮 bal(S) for any collection 𝒮 of subsets of X.
• In any topological vector space, the balanced hull of any open neighborhood of 0 is again open.
• If X is a Hausdorff topological vector space and if K is a compact subset of X, then the balanced hull of K is compact.[5]

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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