Review of Convex, Balanced, Absolutely Convex, and Absorbent Sets
Review of Convex, Balanced, Absolutely Convex, and Absorbent Sets
Let $E$ be a vector space and let $A \subset E$.
Convex Sets
- Recall from the Convex and Balanced Sets of Vectors page that $A$ is Convex if for all $x, y \in A$ and for all $\lambda, \mu \in \mathbf{F}$ with $\lambda, \mu \geq 0$ and $\lambda + \mu = 1$ we have that:
\begin{align} \quad \lambda x + \mu y \in A \end{align}
- From the page mentioned above and the **Properties of Convex Sets of Vectors page, we summarize some properties of convex sets below.
Properties of Convex Sets | |
(1) | If $A$ is convex and $x \in E$ then $x + A$ is convex. |
(2) | If $A$ is convex and $\lambda \in \mathbf{F}$ then $\lambda A$ is convex. |
(3) | If $A$ and $B$ are convex then $A + B$ is convex. |
(4) | If $\{ A_i : i \in I \}$ is a collection of convex sets then $\displaystyle{\bigcap_{i \in I} A_i}$ is convex. |
(5) | If $A$ is convex and $\lambda, \mu \in \mathbb{R}$ then $\lambda A + \mu A = (\lambda + \mu)A$. |
Balanced Sets
- We said that $A$ is Balanced if for all $\lambda \in \mathbf{F}$ with $|\lambda| \leq 1$ we have that $\lambda A \subseteq A$.
- From the page mentioned at the beginning of the page, and from the **Properties of Balanced Sets of Vectors page, we summarize some properties of balanced sets below.
Properties of Balanced Sets | |
(1) | If $A$ is a nonempty balanced set then $o \in A$. |
(2) | If $\{ A_i : i \in I \}$ is a collection of balanced sets then $\displaystyle{\bigcup_{i \in I} A_i}$ is balanced. |
(3) | If $\{ A_i : i \in I \}$ is a collection of balanced sets then $\displaystyle{\bigcap_{i \in I} A_i}$ is balanced. |
(4) | If $A$ and $B$ are balanced sets then $A + B$ is a balanced set. |
(5) | If $A$ is balanced and $\lambda \in \mathbf{F}$ then $\lambda A = |\lambda| A$. |
(6) | If $A$ is balanced and $\lambda, \mu \in \mathbf{F}$ are such that $|\lambda| \leq |\mu|$ then $\lambda A \subseteq \mu A$. |
Absolutely Convex Sets
- On the Absolutely Convex Sets of Vectors page we defined $A$ to be Absolutely Convex if it is both convex and balanced. We summarize some properties of absolutely convex sets below.
Properties of Absolutely Convex Sets | |
(1) | If $A$ and $B$ are absolutely convex then $A + B$ is absolutely convex. |
(2) | If $A$ is absolutely convex and $\lambda \in \mathbf{F}$ then $\lambda A$ is absolutely convex. |
(3) | If $A$ is a nonempty absolutely convex set then $o \in A$. |
(4) | If $A$ is absolutely convex and $\lambda, \mu \in \mathbf{F}$ are such that $|\lambda| \leq |\mu|$ then $\lambda A \subseteq \mu A$. |
Absorbent Sets
- On the Absorbent Sets of Vectors page we defined $A$ to be Absorbent if for every $x \in E$ there exists a $\lambda > 0$ such that if $\mu \in \mathbf{F}$ is such that $|\mu| \geq \lambda$ then:
\begin{align} \quad x \in \mu A \end{align}
- From the page above and the Properties of Absorbent Sets of Vectors page we summarize some properties of absorbent sets below.
Properties of Absorbent Sets | |
(1) | If $\{ A_1, A_2, ..., A_n \}$ is a finite collection of absorbent sets then $\displaystyle{\bigcap_{i=1}^{n} A_i}$ is absorbent. |
(2) | If $A$ is absorbent then for each $x \in E$ there exists a $0 < \mu < 1$ such that $-\mu x \in A$. |