Theorem: Convex Balanced Hull of Finite Union of Convex Balanced Sets

Let $\GF \in \set {\R, \C}$.
Let $X$ be a vector space over $\GF$.
Let $K_1, \ldots, K_n \subseteq X$ be convex balanced sets.
Let $\operatorname {convbal}$ denote convex balanced hull.
Then:
:$\ds \map {\operatorname {convbal} } {\bigcup_{j \mathop = 1}^n K_j} = \set {\sum_{j \mathop = 1}^n \lambda_j x_j : \lambda_1, \ldots, \lambda_n \in \GF, \, x_j \in K_j \text { for all } 1 \le k \le n \text { and } \sum_{j \mathop = 1}^n \cmod {\lambda_j} \le 1}$

Proof:
Let:
:$\ds C = \set {\sum_{j \mathop = 1}^n \lambda_j x_j : \lambda_1, \ldots, \lambda_n \in \GF, \, x_j \in K_j \text { for all } 1 \le k \le n \text { and } \sum_{j \mathop = 1}^n \cmod {\lambda_j} \le 1}$
Let $x, y \in C$ and $t \in \closedint 0 1$.
Then there exists $x_j \in K_j$ ($1 \le j \le n$), $y_j \in K_j$ ($1 \le j \le m$), $\lambda_1, \ldots, \lambda_n, \mu_1, \ldots, \mu_m \in \GF$ such that:
:$\ds x = \sum_{j \mathop = 1}^n \lambda_j x_j$
and:
:$\ds y = \sum_{j \mathop = 1}^m \mu_j y_j$ 
Consider the case $n \ne m$. 
The case $n = m$ follows with no modifications.
{{WLOG}} that $n > m$.
Since each $K_j$ is balanced, we have ${\mathbf 0}_X \in K_j$ for each $j$. 
Hence for $m + 1 \le j \le n$ we can take $y_j = {\mathbf 0}_X$ to have:
:$\ds y = \sum_{j \mathop = 1}^n \mu_j y_j$
with $y_j \in K_j$ for each $1 \le j \le n$. 
Then we have:
:$\ds t x + \paren {1 - t} y = \sum_{j \mathop = 1}^n \paren {t \lambda_j x_j + \paren {1 - t} \mu_j y_j}$
Set:
:$\ds s_j = t \cmod {\lambda_j} + \paren {1 - t} \cmod {\mu_j}$ 
We then have:
:$\ds t x + \paren {1 - t} y = \sum_{j \mathop = 1}^n s_j \paren {\frac {t \cmod {\lambda_j} } {s_j} e^{i \theta_j} x_j + \frac {\paren {1 - t} \cmod {\mu_j} } {s_j} e^{i \phi_j} y_j}$ 
where we take $\theta_j \in \R$ and $\phi_j \in \R$ such that:
:$e^{i \theta_j} \cmod {\lambda_j} = \lambda_j$
and:
:$e^{i \phi_j} \cmod {\mu_j} = \mu_j$
Since $K_j$ is balanced and $\cmod {e^{i \theta_j} } = 1$, $\cmod {e^{i \phi_j} } = 1$, we have:
:$e^{i \theta_j} x_j \in K_j$ and $e^{i \phi_j} y_j \in K_j$ for each $1 \le j \le n$. 
Further, we have:
:$\ds \frac {t \cmod {\lambda_j} } {s_j} + \frac {\paren {1 - t} \cmod {\mu_j} } {s_j} = 1$ 
Hence:
:$\frac {t \cmod {\lambda_j} } {s_j} e^{i \theta_j} x_j + \frac {\paren {1 - t} \cmod {\mu_j} } {s_j} e^{i \phi_j} y_j \in K_j$
since $K_j$ is convex.
Since:
{{begin-eqn}}
{{eqn	| l = \sum_{j \mathop = 1}^n s_j 
		| r = t \sum_{j \mathop = 1}^n \cmod {\lambda_j} + \paren {1 - t} \sum_{j \mathop = 1}^n \cmod {\mu_j} 
}}
{{eqn	| o = \le
		| r = t + \paren {1 - t}
}}
{{eqn	| r = 1
}}
{{end-eqn}}
Hence we have:
:$t x + \paren {1 - t} y \in C$ 
So $C$ is convex.
Now let $x \in C$.
Let $\lambda_1, \ldots, \lambda_n \in \GF$ and $x_j \in K_j$ ($1 \le j \le n$) be such that:
:$\ds \sum_{j \mathop = 1}^n \cmod {\lambda_j} \le 1$ 
and:
:$\ds x = \sum_{j \mathop = 1}^n \lambda_j x_j$ 
Let $\alpha \in \GF$ be such that $\cmod \alpha \le 1$.
Then we have:
:$\ds \sum_{j \mathop = 1}^n \cmod {\alpha \lambda_j} \le \sum_{j \mathop = 1}^n \cmod {\lambda_j} \le 1$ 
and:
:$\ds \alpha x = \sum_{j \mathop = 1}^n \paren {\alpha \lambda_j} x_j$ 
Hence $\alpha x \in C$.
Hence $C$ is balanced.
{{qed}}
Category:Convex Balanced Hulls
Category:Convex Sets (Vector Spaces)
Category:Balanced Sets
bnz5wb154kyf1vqybp0eyqo8rirec9d