Balanced set

In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field ${\displaystyle \mathbb {K} }$ with an absolute value function ${\displaystyle |\cdot |}$) is a set ${\displaystyle S}$ such that ${\displaystyle aS\subseteq S}$ for all scalars ${\displaystyle a}$ satisfying ${\displaystyle |a|\leq 1.}$

The balanced hull or balanced envelope of a set ${\displaystyle S}$ is the smallest balanced set containing ${\displaystyle S.}$ The balanced core of a subset ${\displaystyle S}$ is the largest balanced set contained in ${\displaystyle S.}$

Definition[]

Suppose that ${\displaystyle X}$ is a vector space over the field ${\displaystyle \mathbb {K} }$ of real or complex numbers. Elements of ${\displaystyle \mathbb {K} }$ are called scalars.

Notation: If ${\displaystyle S}$ is a set, ${\displaystyle a}$ is a scalar, and ${\displaystyle B\subseteq \mathbb {K} }$ then let

and for any ${\displaystyle 0\leq r\leq \infty ,}$ let

denote the closed ball (respectively, the open ball) of radius ${\displaystyle r}$ in ${\displaystyle \mathbb {K} }$ centered at ${\displaystyle 0}$ where ${\displaystyle B_{0}=\varnothing ,}$ ${\displaystyle {\overline {B_{0}}}=\{0\},}$ and ${\displaystyle B_{\infty }:={\overline {B_{\infty }}}=\mathbb {K} .}$ Every balanced subset of the field ${\displaystyle \mathbb {K} }$ is of the form ${\displaystyle {\overline {B_{r}}}}$ or ${\displaystyle B_{r}}$ for some ${\displaystyle 0\leq r\leq \infty .}$

A subset ${\displaystyle S}$ of ${\displaystyle X}$ is called a balanced set or balanced if it satisfies any of the following equivalent conditions:

1. Definition: ${\displaystyle aS\subseteq S}$ for all scalars ${\displaystyle a}$ satisfying ${\displaystyle |a|\leq 1.}$
2. ${\displaystyle {\overline {B_{1}}}S\subseteq S,}$ where ${\displaystyle {\overline {B_{1}}}:=\{a\in \mathbb {K} :|a|\leq 1\}.}$
3. ${\displaystyle S={\overline {B_{1}}}S.}$^[1]
4. For every ${\displaystyle s\in S,}$ ${\displaystyle S\cap \mathbb {K} s={\overline {B_{1}}}(S\cap \mathbb {K} s).}$
   • ${\displaystyle \mathbb {K} s=\operatorname {span} \{s\}}$ is a ${\displaystyle 0}$ (if ${\displaystyle s=0}$) or ${\displaystyle 1}$ (if ${\displaystyle s\neq 0}$) dimensional vector subspace of ${\displaystyle X.}$
   • If ${\displaystyle R:=S\cap \mathbb {K} s}$ then the above equality becomes ${\displaystyle R={\overline {B_{1}}}R,}$ which is exactly the previous condition for a set to be balanced. Thus, ${\displaystyle S}$ is balanced if and only if for every ${\displaystyle s\in S,}$ ${\displaystyle A\cap \mathbb {K} s}$ is a balanced set (according to any of the previous defining conditions).
5. For every 1-dimensional vector subspace ${\displaystyle Y}$ of ${\displaystyle \operatorname {span} S,}$ ${\displaystyle S\cap Y}$ is a balanced set (according to any defining condition other than this one).
6. For every ${\displaystyle s\in S,}$ there exists some ${\displaystyle 0\leq r\leq \infty }$ such that ${\displaystyle S\cap \mathbb {K} s=B_{r}s}$ or ${\displaystyle S\cap \mathbb {K} s={\overline {B_{r}}}s.}$

If ${\displaystyle S}$ is a convex set then this list may be extended to include:

7. ${\displaystyle aS\subseteq S}$ for all scalars ${\displaystyle a}$ satisfying ${\displaystyle |a|=1.}$^[2]

If ${\displaystyle \mathbb {K} =\mathbb {R} }$ then this list may be extended to include:

8. ${\displaystyle S}$ is symmetric (meaning ${\displaystyle -S=S}$) and ${\displaystyle [0,1)S\subseteq S.}$

The balanced hull of a subset ${\displaystyle S}$ of ${\displaystyle X,}$ denoted by ${\displaystyle \operatorname {bal} S,}$ is defined in any of the following equivalent ways:

1. Definition: ${\displaystyle \operatorname {bal} S}$ is the smallest (with respect to ${\displaystyle \,\subseteq \,}$) balanced subset of ${\displaystyle X}$ containing ${\displaystyle S.}$
2. ${\displaystyle \operatorname {bal} S}$ is the intersection of all balanced sets containing ${\displaystyle S.}$
3. ${\displaystyle \operatorname {bal} S=\bigcup _{|a|\leq 1}(aS).}$
4. ${\displaystyle \operatorname {bal} S={\overline {B_{1}}}S.}$^[1]

The balanced core of a subset ${\displaystyle S}$ of ${\displaystyle X,}$ denoted by ${\displaystyle \operatorname {balcore} S,}$ is defined in any of the following equivalent ways:

1. Definition: ${\displaystyle \operatorname {balcore} S}$ is the largest (with respect to ${\displaystyle \,\subseteq \,}$) balanced subset of ${\displaystyle S.}$
2. ${\displaystyle \operatorname {balcore} S}$ is the union of all balanced subsets of ${\displaystyle S.}$
3. ${\displaystyle \operatorname {balcore} S=\varnothing }$ if ${\displaystyle 0\not \in S}$ while ${\displaystyle \operatorname {balcore} S=\bigcap _{|a|\geq 1}(aS)}$ if ${\displaystyle 0\in S.}$

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. For an example, let ${\displaystyle X:=\mathbb {R} ^{2}}$ and let the convex subset be ${\displaystyle S:=[-1,1]\times \{1\},}$ which is a horizontal closed line segment lying above the ${\displaystyle x-}$axis. The balanced hull ${\displaystyle \operatorname {bal} S}$ is a non-convex subset that is "hour glass shaped" and equal to the union of two closed and filled isosceles triangles ${\displaystyle T_{1}}$ and ${\displaystyle T_{2},}$ where ${\displaystyle T_{2}=-T_{1}}$ and ${\displaystyle T_{1}}$ is the filled triangle whose vertices are the origin together with the endpoints of ${\displaystyle S}$ (said differently, ${\displaystyle T_{1}}$ is the convex hull of ${\displaystyle S\cup \{(0,0)\}}$ while ${\displaystyle T_{2}}$ is the convex hull of ${\displaystyle (-S)\cup \{(0,0)\}}$).
• 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 the origin is again balanced.

Examples

• If ${\displaystyle S\subseteq X}$ is any subset and ${\displaystyle B_{1}:=\left\{a\in \mathbb {K} :|a|<1\right\}}$ then ${\displaystyle B_{1}S}$ is a balanced set.
  • In particular, if ${\displaystyle U\subseteq X}$ is any balanced neighborhood of the origin in a TVS ${\displaystyle X}$ then ${\displaystyle \operatorname {Int} U=B_{1}U=\bigcup _{0<|a|<1}aU\subseteq U.}$
• If ${\displaystyle \mathbb {K} }$ is the field real or complex numbers and ${\displaystyle X=\mathbb {K} }$ is the normed space over ${\displaystyle \mathbb {K} }$ with the usual Euclidean norm, then the balanced subsets of ${\displaystyle X}$ are exactly the following:^[4]
  1. ${\displaystyle \varnothing }$
  2. ${\displaystyle X}$
  3. ${\displaystyle \{0\}}$
  4. ${\displaystyle \left\{x\in X:|x|<r\right\}}$ for some real ${\displaystyle r>0}$
  5. ${\displaystyle \left\{x\in X:|x|\leq r\right\}}$ for some real ${\displaystyle r>0.}$
• 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 ${\displaystyle X=\mathbb {R} ^{2}}$ (${\displaystyle X}$ is a vector space over ${\displaystyle \mathbb {R} }$), ${\displaystyle {\overline {B_{1}}}}$ is the closed unit ball in ${\displaystyle X}$ centered at the origin, ${\displaystyle x_{0}\in X}$ is non-zero, and ${\displaystyle L:=\mathbb {R} x_{0},}$ then the set ${\displaystyle R:={\overline {B_{1}}}\cup L}$ is a closed, symmetric, and balanced neighborhood of the origin in ${\displaystyle X.}$ More generally, if ${\displaystyle C}$ is any closed subset of ${\displaystyle X}$ such that ${\displaystyle (0,1)C\subseteq C,}$ then ${\displaystyle S:={\overline {B_{1}}}\cup C\cup (-C)}$ is a closed, symmetric, and balanced neighborhood of the origin in ${\displaystyle X.}$ This example can be generalized to ${\displaystyle \mathbb {R} ^{n}}$ for any integer ${\displaystyle n\geq 1.}$
• The Cartesian product of a family of balanced sets is balanced in the product space of the corresponding vector spaces (over the same field ${\displaystyle \mathbb {K} }$).
• Consider ${\displaystyle \mathbb {C} ,}$ the field of complex numbers, as a 1-dimensional vector space. The balanced sets are ${\displaystyle \mathbb {C} }$ 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, ${\displaystyle \mathbb {C} }$ and ${\displaystyle \mathbb {R} ^{2}}$ are entirely different as far as scalar multiplication is concerned.
• If ${\displaystyle p}$ is a seminorm on a linear space ${\displaystyle X,}$ then for any constant ${\displaystyle c>0}$ the set ${\displaystyle \left\{x\in X:p(x)\leq c\right\}}$ is balanced.
• Let ${\displaystyle X=\mathbb {R} ^{2}}$ and let ${\displaystyle B}$ be the union of the line segment between ${\displaystyle (-1,0)}$ and ${\displaystyle (1,0)}$ and the line segment between ${\displaystyle (0,-1)}$ and ${\displaystyle (0,1).}$ Then ${\displaystyle B}$ is balanced but not convex or absorbing. However, ${\displaystyle \operatorname {span} B=X.}$
• Let ${\displaystyle X=\mathbb {R} ^{2}}$ and for every ${\displaystyle 0\leq t\leq \pi ,}$ let ${\displaystyle r_{t}}$ be any positive real number and let ${\displaystyle B^{t}}$ be the (open or closed) line segment between the points ${\displaystyle (\cos t,\sin t)}$ and ${\displaystyle -(\cos t,\sin t).}$ Then the set ${\displaystyle B=\bigcup _{0\leq t<\pi }r_{t}B^{t}}$ 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 ${\displaystyle xy=1}$ in${\displaystyle X=\mathbb {R} ^{2}.}$

Properties[]

Properties of balanced sets

• A set is absolutely convex if and only if it is convex and balanced.
• If ${\displaystyle B}$ is balanced then for any scalar ${\displaystyle a,}$ ${\displaystyle aB=|a|B.}$
• If ${\displaystyle B}$ is balanced then for any scalars ${\displaystyle a}$ and ${\displaystyle b}$ such that ${\displaystyle |a|\leq |b|,}$ ${\displaystyle aB\subseteq bB.}$
• The union of ${\displaystyle \{0\}}$ and the interior of a balanced set is balanced.
• If ${\displaystyle B}$ is a balanced subset of ${\displaystyle X,}$ then ${\displaystyle B}$ is absorbing in ${\displaystyle X}$ if and only if for all ${\displaystyle x\in X,}$ there exists ${\displaystyle r>0}$ such that ${\displaystyle x\in rB.}$^[2]
• If ${\displaystyle B}$ is a balanced subset of ${\displaystyle X,}$ then ${\displaystyle B}$ is absorbing in ${\displaystyle \operatorname {span} B.}$
• The Minkowski sum of two balanced sets is balanced.
• Every balanced set is symmetric.
• Every balanced set is star-shaped (at 0).
• Suppose ${\displaystyle B}$ is balanced. If ${\displaystyle Y}$ is a 1-dimensional vector subspace of ${\displaystyle X}$ then ${\displaystyle B\cap Y}$} is convex and balanced. If ${\displaystyle Y}$ is a 1-dimensional vector subspace of ${\displaystyle \operatorname {span} B}$ then ${\displaystyle B\cap Y}$} is also absorbing in ${\displaystyle Y.}$
• If ${\displaystyle B\neq \varnothing }$ is a balanced then for any ${\displaystyle x\in X,}$ ${\displaystyle B\cap \mathbb {R} x}$ is a convex balanced set containing the origin. If ${\displaystyle B}$ is a neighborhood of ${\displaystyle 0}$ in ${\displaystyle X}$ then ${\displaystyle B\cap \mathbb {R} x}$ is a convex balanced neighborhood of ${\displaystyle 0}$ in the real vector subspace ${\displaystyle \mathbb {R} x.}$

Properties of balanced hulls

• ${\displaystyle a\operatorname {bal} S=\operatorname {bal} (aS)}$ for any subset ${\displaystyle S}$ of ${\displaystyle X}$ and any scalar ${\displaystyle a.}$
• ${\displaystyle \operatorname {bal} \left(\bigcup _{S\in {\mathcal {S}}}S\right)=\bigcup _{S\in {\mathcal {S}}}\operatorname {bal} S}$ for any collection ${\displaystyle {\mathcal {S}}}$ of subsets of ${\displaystyle X.}$
• In any topological vector space, the balanced hull of any open neighborhood of the origin is again open.
• If ${\displaystyle X}$ is a Hausdorff topological vector space and if ${\displaystyle K}$ is a compact subset of ${\displaystyle X,}$ then the balanced hull of ${\displaystyle K}$ is compact.^[5]

Balanced core

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

See also[]

• 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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