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 $0<p\le 1$, then $M$ is called $p$-convex if $M$ fulfills the following property:

${\mathrm{\forall }}_{x,y\in M;\lambda ,\mu \ge 0}:{\lambda }^p+{\mu }^p=1⟹\lambda x+\mu y\in M$

Definition: absolute p-convex

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

${\mathrm{\forall }}_{x,y\in M}:|\lambda {|}^p+|\mu {|}^p\le 1⟹\lambda x+\mu y\in M$

Definition: p-convex hull

The $p$-convex hull of the set $M$ (label: ${𝒞}_p(M)$) is the intersection over all $p$-convex sets containing $M$.

${𝒞}_p(M):=\bigcap _{\stackrel{\tilde{M}\supseteq M}{\tilde{M}p-convex}}\tilde{M}$

Definition: absolute p-convex hull

The absolutely $p$-convex hull of the set $M$ (label: ${\mathrm{\Gamma }}_p(M)$) is the section over all absolutely $p$-convex sets containing $M$.

${\mathrm{\Gamma }}_p(M):=\bigcap _{\stackrel{\tilde{M}\supseteq M}{\tilde{M}absolutep-convex}}\tilde{M}$

Lemma: Display of the absolutely p-convex hull

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

$${\mathrm{\Gamma }}_p(M)=\{\sum _{j=1}^n{\alpha }_j{x}_j:n\in \mathbb{N}\wedge x_j\in M\wedge \sum _{j=1}^n|{\alpha }_j{|}^p\le 1\}=:\hat{M}$$

Proof

3 subassertions are shown, where (1) and (2) gives ${\mathrm{\Gamma }}_p(M)\subseteq \hat{M}$ and (3) gives the subset relation $\hat{M}\subseteq {\mathrm{\Gamma }}_p(M)$.

• (Proof part 1) $M\subset \hat{M}$,
• (Proof part 2) $\hat{M}$ is absolutely $p$-convex and.
• (Proof part 3) $\hat{M}$ is contained in any absolutely $p$-convex set $\tilde{M}\supset M$.

Proof part 1

$M\subset \hat{M}$, because $M=\{{\alpha }_{x}x:{\alpha }_x=1\wedge x\in M\}\subset \hat{M}$

Proof part 2

Now let $x,y\in \hat{M}$ and $|\alpha {|}^p+|\beta {|}^p\le 1$ be given. One must show that $\alpha x+\beta y\in \hat{M}$.

Proof Part 2.1 - Absolute p-convex

Let $x,y\in \hat{M}$ now have $x,y\in \hat{M}$ the following representations:

• $x=\sum _{i=1}^m{\alpha }_i{x}_i$ with $\sum _{i=1}^m|{\alpha }_i{|}^p\le 1$
• $y=\sum _{i=1}^n{\beta }_i{y}_i$ with $\sum _{i=1}^n|{\beta }_i{|}^p\le 1$.

Now we have to show that the absolute $p$-convex combination is an element of $\hat{M}$, i.e. $\alpha x+\beta y\in \hat{M}$

proof-part-2.2-absolutely-p-convex

$\hat{M}$ is absolutely $p$-convex, because it holds with $|\alpha {|}^p+|\beta {|}^p\le 1$:

$\begin{array}{ccc}\sum _{i=1}^m|\alpha {\alpha }_i{|}^p+\sum _{j=1}^n|\beta {\beta }_j{|}^p& =& |\alpha {|}^p\underset{\le 1}{\underbrace{\sum _{i=1}^m|{\alpha }_i{|}^p}}+|\beta {|}^p\underset{\le 1}{\underbrace{\sum _{j=1}^n|{\beta }_j{|}^p}}\\ & \le & |\alpha {|}^p+|\beta {|}^p\le 1.\end{array}$

This gives:

$$\alpha \cdot x+\beta \cdot y=.\alpha \sum _{i=1}^m{\alpha }_i{x}_i+\beta \sum _{j=1}^n{\beta }_j{y}_j\in \hat{M}.$$

Proof Part 2.3 - Zero Vector

$0_V\in \hat{M}$, because it holds $0_V=\alpha \cdot x$ with $\alpha =0=|\alpha {|}^p\le 1$ and any $x\in M$ gets $0_V=\alpha \cdot x\in \hat{M}$.

Proof part 3

We now show that the absolutely $p$-convex hull is contained in every absolutely $p$-convex superset $\tilde{M}$ 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

$$\sum _{j=1}^n{\alpha }_j{x}_j\text{ with }{x}_j\in M\text{ and }\sum _{j=1}^n|{\alpha }_j{|}^p\le 1$$

in a given absolutely $p$-convex set $\tilde{M}\supset M$ is contained.

Proof Part 3.2 - Induction Start

For $n=2$, the assertion follows via the definition of an absolutely $p$-convex set $\tilde{M}\supset M$.

Proof Part 3.3 - Induction Precondition

Now let the condition for $n$ hold, i.e.:

$$\sum _{j=1}^n{\alpha }_j{x}_j\in \tilde{M}\text{ with }{x}_j\in M\text{ and }\sum _{j=1}^n|{\alpha }_j{|}^p\le 1.$$

Proof Part 3.4 - Induction Step

For $n+1$, the assertion follows as follows:

Let $x:=\sum _{j=1}^{n+1}{\alpha }_j{x}_j$ and $\sum _{j=1}^{n+1}|{\alpha }_j{|}^p\le 1$ with $x_j\in M$ for all $j\in \{1,\dots ,n+1\}$. $x\in \tilde{M}$ 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 $j\in \{1,\dots ,n\}$ .

Proof Part 3.6 - Constructing a p-convex combination of n summands

We now construct a sum of non-negative summands ${\beta }_j\ge 0$

$${\beta }_j:=\frac{{\alpha }_j}{\sqrt[p]{1-|{\alpha }_{n+1}{|}^p}}\text{ with }\sum _{j=1}^n{\left|{\beta }_j\right|}^p\le 1$$

Proof part 3.7 - Application of the induction assumption

So let $|{\alpha }_{n+1}|<1$. The inequality

$$\sum _{j=1}^n\underset{=|{\beta }_j{|}^p}{\underbrace{{\left|\frac{{\alpha }_j}{\sqrt[p]{1-|{\alpha }_{n+1}{|}^p}}\right|}^p}}=\frac{1}{1-|{\alpha }_{n+1}{|}^p}\cdot \underset{\le 1-|{\alpha }_{n+1}{|}^p}{\underbrace{\sum _{j=1}^n|{\alpha }_j{|}^p}}\le 1$$

Returns after induction assumption $z:=.\sum _{j=1}^n{\beta }_j\cdot x_j=.\sum _{j=1}^n\frac{{\alpha }_j}{\sqrt[p]{1-|{\alpha }_{n+1}{|}^p}}\cdot x_j\in \tilde{M}$.

Proof Part 3.8 - Induction Step

Since $\tilde{M}$ is absolutely $p$-convex, it follows with ${\left(\sqrt[p]{1-|{\alpha }_{n+1}{|}^p}\right)}^p+|{\alpha }_{n+1}{|}^p=1$

$$\tilde{M}\ni \left(\sqrt[p]{1-|{\alpha }_{n+1}{|}^p}\right)z+{\alpha }_{n+1}{x}_{n+1}=\sum _{j=1}^n{\alpha }_j{x}_j+{\alpha }_{n+1}{x}_{n+1}=.\sum _{j=1}^{n+1}{\alpha }_j{x}_j.$$

Proof 4

From the proof parts $(1)$, $(2)$ and $(3)$ together the assertion follows. $◻$

Lemma: p-convex hull

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

$${𝒞}_p(M)=\{.\sum _{j=1}^n{\alpha }_j{x}_j:n\in \mathbb{N}\wedge x_j\in M\wedge {\alpha }_j\in [0,1]\wedge \sum _{j=1}^n{\alpha }_j^p=1\}$$

Proof: task for learners

Transfer the above proof analogously to the $p$-convex hull.

See also

• Linear algebra
• Minkowski functional
• P-regularity
• PC-regularity
• Convex Combination
• Lemma - subadditivity p-convexity

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