P-convex hull

Introduction
For $$p$$-norms are a generalization of norms. The definition requires the notion of (absolute) $$p$$-convex hull (see Köthe 1966 ).

Definition: p-convex
Let $M$ be a subset of a vector space $V$  and $0< p \leq 1$, then $M$  is called $p$ -convex if $M$  fulfills the following property:

\forall_{\displaystyle x,y\in M;\lambda,\mu\geq 0} : \lambda^p+\mu^p = 1 \,\Longrightarrow\, \lambda x + \mu y \in M     $$

Definition: absolute p-convex
Let $M$ be a subset of a vector space $V$  and $0< p \leq 1$, then $M$  is said to be absolutely $p$ -convex if $M$  fulfills the following property:

\forall_{\displaystyle x,y\in M} : |\lambda|^p+|\mu|^p \leq 1 \,\Longrightarrow\, \lambda x + \mu y \in M     $$

Definition: p-convex hull
The $p$ -convex hull of the set $M$ (label: $\mathcal{C}_p(M)$ ) is the intersection over all $p$ -convex sets containing $M$.
 * $$\mathcal{C}_p(M) := \displaystyle \bigcap_{\stackrel{\widetilde{M} \supseteq M}{\widetilde{M}\, p-convex} } \widetilde{M} $$

Definition: absolute p-convex hull
The absolutely $p$ -convex hull of the set $M$ (label: $\Gamma_p(M)$ ) is the section over all absolutely $p$ -convex sets containing $M$.
 * $$\Gamma_p(M) := \displaystyle \bigcap_{\stackrel{\widetilde{M} \supseteq M}{\widetilde{M}\, absolute \, p-convex} } \widetilde{M} $$

Lemma: Display of the absolutely p-convex hull
Let $M$ be a subset of a vector space $V$  over the body $\mathbb{K}$  and $0< p \leq 1$, then the absolute $p$ -convex hull of $M$  can be written as follows:

\Gamma_p(M)=\left\{ \sum_{j=1}^{n} \alpha_j x_j \, : \, n \in \mathbb{N} \wedge x_j\in M       \wedge \sum_{j=1}^{n} |\alpha_j|^p \leq 1 \right\} =: \widehat{M} $$

Proof
3 subassertions are shown, where (1) and (2) gives $$\Gamma_p(M) \subseteq \widehat{M}$$ and (3) gives the subset relation $$ \widehat{M} \subseteq \Gamma_p(M)$$.
 * (Proof part 1) $M\subset \widehat{M}$ ,
 * (Proof part 2) $\widehat{M}$ is absolutely $p$ -convex and.
 * (Proof part 3) $\widehat{M}$ is contained in any absolutely $p$ -convex set $\widetilde{M}\supset M$.

Proof part 1
$M\subset \widehat{M}$, because $M=\{\alpha_x x: \alpha_x=1 \wedge x\in M\} \subset \widehat{M}$

Proof part 2
Now let $x,y \in \widehat{M}$ and $ |\alpha|^p + |\beta|^p\leq 1 $  be given. One must show that $ \alpha x + \beta y \in \widehat{M} $.

Proof Part 2.1 - Absolute p-convex
Let $x,y \in \widehat{M}$ now have $x,y \in \widehat{M}$  the following representations: Now we have to show that the absolute $$p$$-convex combination is an element of $$\widehat{M}$$, i.e. $\alpha x + \beta y \in \widehat{M}$
 * $$ \displaystyle x = \sum_{i=1}^{m} \alpha_i x_i $$ with $$ \displaystyle \sum_{i=1}^{m} |\alpha_i|^p \leq 1 $$
 * $$ \displaystyle y = \sum_{i=1}^{n} \beta_i y_i $$ with $$ \displaystyle \sum_{i=1}^{n} |\beta_i|^p \leq 1 $$.

proof-part-2.2-absolutely-p-convex
$$\widehat{M}$$ is absolutely $p$ -convex, because it holds with $ |\alpha|^p + |\beta|^p\leq 1 $ :

\begin{array}{rcl} \sum_{i=1}^{m} |\alpha\alpha_i|^p + \sum_{j=1}^{n} |\beta\beta_j|^p &=&               |\alpha|^p\underbrace{\sum_{i=1}^{m} |\alpha_i|^p}_{\leq 1} + |\beta|^p \underbrace{\sum_{j=1}^{n} |\beta_j|^p}_{\leq 1} \\          &\leq & |\alpha|^p + |\beta|^p \leq 1. \\    \end{array} $$  This gives:

\alpha \cdot x + \beta \cdot y =. \alpha\sum_{i=1}^{m} \alpha_i x_i+\beta \sum_{j=1}^{n} \beta_j y_j \in \widehat{M}. $$

Proof Part 2.3 - Zero Vector
$0_V\in \widehat{M}$, because it holds $0_V=\alpha \cdot x$ with $\alpha = 0 = |\alpha|^p \leq 1$  and any $x\in M$  gets $0_V=\alpha \cdot x \in \widehat{M}$.

Proof part 3
We now show that the absolutely $$p$$-convex hull is contained in every absolutely $$p$$-convex superset $$\widetilde{M}$$ of $$M$$.

Proof Part 3.1 - Induction over Number of Summands
Now let us show inductively via the number of summands $$n$$ that every element of the form

\sum_{j=1}^{n} \alpha_j x_j \mbox{ with } x_j\in M    \mbox{ and } \sum_{j=1}^{n} |\alpha_j|^p \leq 1 $$ in a given absolutely $p$ -convex set $\widetilde{M} \supset M$ is contained.

Proof Part 3.2 - Induction Start
For $n=2$, the assertion follows via the definition of an absolutely $p$ -convex set $\widetilde{M}\supset M$.

Proof Part 3.3 - Induction Precondition
Now let the condition for $n$ hold, i.e.:

\sum_{j=1}^{n} \alpha_j x_j \in \widetilde{M} \mbox{ with } x_j\in M    \mbox{ and } \sum_{j=1}^{n} |\alpha_j|^p \leq 1. $$

Proof Part 3.4 - Induction Step
For $n+1$, the assertion follows as follows:

Let $\displaystyle x:=\sum_{j=1}^{n+1} \alpha_j x_j$ and $\displaystyle \sum_{j=1}^{n+1} |\alpha_j |^p\leq 1$  with $x_j\in M$  for all $j \in \{1,\dots ,n+1\}$. $x\in \widetilde{M}$ is now to be proved.

Proof Part 3.5 - Induction Step
If $\alpha_{n+1}=1$, then there is nothing to show, since then all $$|\alpha_j|=0$$ are for $$ j \in \{1,\ldots , n\}$$.

Proof Part 3.6 - Constructing a p-convex combination of n summands
We now construct a sum of non-negative summands $$ \beta_j \geq 0 $$

\beta_j := \frac{\alpha_j}{ \sqrt[p]{ 1-|\alpha_{n+1}|^p } } \mbox{ with } \sum_{j=1}^{n} \left| \beta_j \right|^p \leq 1 $$

Proof part 3.7 - Application of the induction assumption
So let $|\alpha_{n+1}|<1$. The inequality

\sum_{j=1}^{n} \underbrace{\left| \frac{\alpha_j}{ \sqrt[p]{ 1-|\alpha_{n+1}|^p } } \right|^p}_{= | \beta_j |^p} =    \frac{1}{ 1-|\alpha_{n+1}|^p }  \cdot \underbrace{ \sum_{j=1}^{n} |\alpha_j|^p }_{\leq 1-|\alpha_{n+1}|^p} \leq 1 $$ Returns after induction assumption $\displaystyle z:=. \sum_{j=1}^{n} \beta_j \cdot x_j =. \sum_{j=1}^{n} \frac{\alpha_j}{ \sqrt[p]{ 1-|\alpha_{n+1}|^p } } \cdot x_j \in \widetilde{M}$.

Proof Part 3.8 - Induction Step
Since $\widetilde{M}$ is absolutely $p$ -convex, it follows with $\left(\sqrt[p]{1-|\alpha_{n+1}|^p}\right)^p +|\alpha_{n+1}|^p=1$



\widetilde{M} \ni \left(\sqrt[p]{1-|\alpha_{n+1}|^p}\right) z + \alpha_{n+1}x_{n+1} =  \sum_{j=1}^{n}   \alpha_j x_j + \alpha_{n+1}x_{n+1} =. \sum_{j=1}^{n+1} \alpha_j x_j. $$

Proof 4
From the proof parts $(1)$, $(2)$ and $(3)$  together the assertion follows. $$ \Box $$

Lemma: p-convex hull
Let $M$ be a subset of a vector space $V$  over the body $\mathbb{K}$  and $0< p \leq 1$, then the $p$ -convex hull of $M$  can be written as follows:

{\cal C}_p(M)=\left\{. \sum_{j=1}^{n} \alpha_j x_j \, : \, n \in \mathbb{N} \wedge x_j\in M     \wedge \alpha_j \in [0,1] \wedge \sum_{j=1}^{n} \alpha_j^p = 1 \right\} $$

Proof: task for learners
Transfer the above proof analogously to the $$p$$-convex hull.

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