# Quantitative Hahn-Banach Theorems and Isometric Extensions for Wavelet and Other Banach Spaces

## Abstract

**:**

## 1. Introduction

## 2. Definitions, Designations, and Basic Properties

**Definition 2.1.**For $p\in {(0,\infty )}^{n}$, $q\in (0,\infty )$ and $n\in \mathbb{N}$, let $l{t}_{p,q}=l{t}_{p,q}({\mathbb{Z}}^{n}\times J)$ be the quasi-Banach space of the sequences ${\left\{{t}_{i,j}\right\}}_{i\in {\mathbb{Z}}^{n}}^{j\in J}$ with $J\in \{{\mathbb{N}}_{0},\mathbb{Z}\}$ endowed with the (quasi)norm,

**Remark 2.1.**There are many books and articles dedicated to the wavelet decompositions (and their predecessors) and related characterisations of the function spaces, including [19,21,22,23,24,25,26,27,28,29]. The construction of $l{t}_{p,q}$ above is compatible not only with the known results but also with a more optimal version [20]. Thus, Besov ${B}_{p,q}^{s}{\left({\mathbb{R}}^{n}\right)}_{w}$ and Lizorkin-Triebel ${L}_{p,q}^{s}{\left({\mathbb{R}}^{n}\right)}_{w}$ sequence spaces, which we shall call Besov and Lizorkin-Triebel spaces with wavelet norms, can be defined in various constructive ways (we omit the lengthy details) but the following property always holds. Indeed, throughout the article we only need to know the immediate corollary of the definition(s) that ${B}_{p,q}^{s}{\left({\mathbb{R}}^{n}\right)}_{w}$ is, in fact, isometric to ${l}_{q}\left({\mathbb{N}}_{0},{l}_{p}\left({\mathbb{Z}}^{n}\right)\right)$, while ${L}_{p,q}^{s}{\left({\mathbb{R}}^{n}\right)}_{w}$ is isometric to a 1-complemented subspace of $l{t}_{p,q}\left({\mathbb{Z}}^{n}\times \mathbb{N},{l}_{q}\left({I}_{M}\right)\right)$ for certain $M\in \mathbb{N}$ and contains, in turn, an isometric and 1-complemented copy of $l{t}_{p,q}=l{t}_{p,q}({\mathbb{Z}}^{n}\times {\mathbb{N}}_{0})$.

**Definition 2.2.**For $r\in [1,\infty ]$, a finite, or countable set I and a set of quasi-Banach spaces ${\left\{{X}_{i}\right\}}_{i\in I}$, let its ${l}_{r}$-sum ${l}_{r}(I,{\left\{{A}_{i}\right\}}_{i\in I})$ be the space of the sequences $x={\left\{{x}_{i}\right\}}_{i\in I}\in {\prod}_{i\in I}{X}_{i}$ with the finite norm

#### 2.1. Independently Generated Spaces, $l{t}_{p,q}$ and $l{t}_{p,q}^{*}$

**Lemma 2.1.**Let X be a Banach space, and $P\in \mathcal{L}\left(X\right)$ be a projector onto its subspace $Y\subset X$. Assume also that ${Q}_{Y}:\phantom{\rule{3.33333pt}{0ex}}X\to \tilde{X}=X/\mathrm{Ker}\phantom{\rule{0.166667em}{0ex}}P$ is the quotient map. Then we have

**Lemma 2.2.**Let $p\in {[1,\infty )}^{n}$ and $q\in [1,\infty )$. Then the spaces $l{t}_{p,q}\left({\mathbb{R}}^{n}\right)$ and $l{t}_{{p}^{\prime},{q}^{\prime}}{\left({\mathbb{R}}^{n}\right)}^{*}$ contain isometric 1-complemented copies of ${l}_{p}\left({\mathbb{N}}^{n}\right)$, ${l}_{q}\left(\mathbb{N}\right)$ and ${l}_{p}\left({\mathbb{Z}}^{n},{l}_{q}\left(\mathbb{N}\right)\right)$, and the spaces $l{t}_{p,q}\left(F\right)$(see Definition $2.1)$ and $l{t}_{{p}^{\prime},{q}^{\prime}}{\left(F\right)}^{*}$ contain isometric 1-complemented copies of ${l}_{q}\left(\mathbb{N}\right)$, ${l}_{p}\left({I}_{m},{l}_{q}\left(\mathbb{N}\right)\right)$ for every $m\in {\mathbb{N}}^{n}$.

**Remark 2.2.**There are several known ways of detecting copies of ${l}_{\sigma}$-spaces in a Banach space. It seems that, at least, in the setting of a regular domain G one can uncover almost isometric copies in Besov and Lizorkin-Triebel spaces with the aid of the remarkable results due to M. Levy [31], M. Mastylo [32] and V. Milman [33] (in combination with the results in Section 10 in [2]). Direct constructions of the isomorphic (and often complemented) copies of various sequence spaces (for example, mixed ${l}_{p}$ and Lorentz ${L}_{p,r}$) in various anisotropic Besov, Sobolev and Lizorkin-Triebel spaces of functions defined on open subsets of ${\mathbb{R}}^{n}$ are presented in Section 3 in [2].

**Definition 2.3. Independently generated spaces**

**Remark 2.3.**

**Lemma 2.3.**Let $X\in {IG}_{+}$$(X\in IG)$. Then there exists $X\in {IG}_{0+}$$(X\in {IG}_{0})$ with $I\left(X\right)=I\left(Y\right)$ and $T\left(X\right)\subset T\left(Y\right)$, such that X is a complemented subspace of a quotient $Y/Z$, where Z is a complemented subspace of Y. Moreover, if $l{t}_{p,q}^{*}\notin T\left(X\right)$, then X is a complemented subspace of Y.

#### 2.2. Non-Commutative Spaces

**Definition 2.4.**Let $\mathcal{M}$ be a von Neumann algebra, and ${\mathcal{M}}_{+}$ be its positive part (cone). A trace on $\mathcal{M}$ is a map $\tau :\phantom{\rule{3.33333pt}{0ex}}{\mathcal{M}}_{+}\to [0,\infty ]$ satisfying

**Definition 2.5.**Let $\mathcal{M}$ be a semifinite algebra, and $x\in {\mathcal{M}}_{+}$. The support $\mathrm{supp}\phantom{\rule{3.33333pt}{0ex}}x$ is the least projection in $\mathcal{M}$ satisfying $px=x$ (or, equivalently, $xp=x$). Assume also that $\mathcal{S}$ is the linear span of the set ${\mathcal{S}}_{+}$ of all $x\in {\mathcal{M}}_{+}$ with $\tau \left(\mathrm{supp}\phantom{\rule{3.33333pt}{0ex}}x\right)<\infty $. For $p\in (0,\infty )$ and $x\in \mathcal{S}$, let the $min(p,1)$-norm of x be defined by

**Examples 2.1**([36]). Let ${H}_{1},{H}_{2}$ be Hilbert spaces and $p\in [1,\infty )$. The Schatten-von Neumann class ${S}_{p}={S}_{p}({H}_{1},{H}_{2})$ is the Banach space of all compact operators $A\in \mathcal{L}({H}_{1},{H}_{2})$ with the finite norm

**Theorem 2.1**([37,38]). Let $\mathcal{M}$ be a semifinite von Neumann algebra, ${p}_{0},{p}_{1}\in [1,\infty ]$, $\theta \in (0,1)$ and $1/p=(1-\theta )/{p}_{0}+\theta /{p}_{1}$. Then

**Theorem 2.2**(Haagerup, see [35]) Let $\mathcal{M}$ be a von Neumann algebra with a normal semifinite faithful weight ϕ, $p\in (0,\infty ))$, and let ${\Lambda}_{p}(\mathcal{M},\varphi )$ be the associated Haagerup ${L}_{p}$-space. Then there are a $min(p,1)$-Banach space X, a directed family ${\left\{({\mathcal{M}}_{i},{\tau}_{i})\right\}}_{i\in I}$ of finite von Neumann algebras and a family ${\left\{{J}_{i}\right\}}_{i\in I}$ of isometric embeddings ${J}_{i}:\phantom{\rule{3.33333pt}{0ex}}{L}_{p}({\mathcal{M}}_{i},{\tau}_{i})$ satisfying

**Theorem 2.3**([39], Corollary 7.7 in in [35]). Let $\mathcal{M}$ be a von Neumann algebra with a normal semifinite faithful weight ϕ and $p\in (0,\infty ))$. Then ${L}_{p}(\mathcal{M},\varphi )$ is a UMD-space (see Definition $7.3)$.

**Lemma 2.4.**For $p\in [1,\infty ]$, the space ${S}_{p}$ contains 1-complemented isometric copies of ${S}_{p}^{n}$, ${l}_{p}\left({I}_{n}\right)$ and ${l}_{p}$ for $n\in \mathbb{N}$. Moreover, an infinite-dimensional ${L}_{p}\left(\mathcal{M}\right)$ contains a 1-complemented isometric copy of ${l}_{p}$.

## 3. Connection between Function and Independently Generated Spaces

**Remark 3.1.**As described in this section, the results of this article in the setting of $IG$-spaces imply the corresponding counterparts in the settings of various types of Besov, Lizorkin-Triebel and Sobolev spaces defined in terms of differences, local polynomial approximations, systems of closed operators and wavelet decompositions, except for the matter of sharpness. The latter can be treated by means of real/harmonic analysis tools and will be presented elsewhere.

## 4. Key Tool: Interpolation of Banach Spaces and Their Subspaces

**Theorem 4.1**

**Theorem 4.2.**([8,49]) For $p\in [1,\infty ]$, $\theta \in (0,1)$, let $({A}_{0},{A}_{1})$ be a compatible couple of Banach spaces, and B be a complemented subspace of ${A}_{0}+{A}_{2}$, whose projector $P\in \mathcal{L}\left({A}_{0}\right)\cap \mathcal{L}\left({A}_{1}\right)$. Then $({A}_{0}\cap B,{A}_{1}\cap B)$ is also compatible, and we have the isomorphisms

**Corollary 4.1.**Let $({A}_{0},{A}_{1})$ be a compatible pair (see Remark $2.3,\phantom{\rule{4pt}{0ex}}\left(d\right))$ of $I{G}_{+}$-spaces $({A}_{0},{A}_{1}\in I{G}_{+})$ with the parameter functions ${p}_{0}$ and ${p}_{1}$ taking values in $[1,\infty ]$ and $\theta \in (0,1)$. Assume also that $1/{p}_{\theta}=(1-\theta )/{p}_{0}+\theta /{p}_{1}$, and that the space ${A}_{\theta}\in I{G}_{+}$ defined by the parameter function ${p}_{\theta}$ is the space with the same tree type as both ${A}_{0}$ and ${A}_{1}$. Then we have the isomorphism ${({A}_{0},{A}_{1})}_{\left[\theta \right]}\asymp {A}_{\theta}$ that becomes the isometry ${({A}_{0},{A}_{1})}_{\left[\theta \right]}={A}_{\theta}$ if $({A}_{0},{A}_{1}\in I{G}_{0+})$.

## 5. From Jacobi, Clarkson and Pichugov Classes to Quantitative Hahn-Banach Theorems

#### 5.1. Counterparts of Jacobi and Parallelogram Identities

**Definition 5.1.**Let X be a Banach space, $\sigma ,{\sigma}_{0},{\sigma}_{1}\in [1,\infty ]$ and ${c}_{J+},{c}_{J}>0$. We say that the space X belongs to the Jacobi class $J({\sigma}_{0},{\sigma}_{1},{c}_{J})$, or the adjoint Jacobi class ${J}_{+}({\sigma}_{0},{\sigma}_{1},{c}_{J+})$, if, for every X-valued stochastic variables ξ and η that are independent and identically distributed on a probability measure space $(\Omega ,\Xi ,p)$, one has, respectively, the estimates

**Remark 5.1.**

**Lemma 5.1.**Let $\sigma \in [1,\infty ]$. Then one has $\mathrm{Im}\phantom{\rule{0.166667em}{0ex}}{D}^{*}D=\mathrm{Im}\phantom{\rule{0.166667em}{0ex}}{D}^{*}=\mathrm{Ker}\phantom{\rule{0.166667em}{0ex}}\mathrm{E}$, $\mathrm{Im}\phantom{\rule{0.166667em}{0ex}}D{D}^{*}=\mathrm{Im}\phantom{\rule{0.166667em}{0ex}}D$, ${2}^{-1}{D}^{*}D=I-\mathrm{E}$, and $P={2}^{-1}D{D}^{*}$ is a bounded projector with $\parallel P|{\mathcal{L}}_{\sigma}(\Omega \times \Omega ,X)\parallel \le 2$. If either μ is purely atomic, or X and ${X}^{*}$ possess the Radon-Nikodým property and $\sigma \in (1,\infty )$, then the operator ${D}^{*}$ is the dual of $D\in \mathcal{L}({L}_{\sigma}(\Omega ,X),{L}_{\sigma}(\Omega \times \Omega ,X))$.

**Corollary 5.1.**Let $(\Omega ,\mu )$ be a probability measure space with either μ being purely atomic uniform measure on the finite discrete Ω$\left(\right|\Omega |<\infty )$, or μ being uniform (continuous) on Ω, and let $\overline{A}=({A}_{0},{A}_{1})$ be a compatible pair of Banach spaces. Assume also that ${u}_{i}:\phantom{\rule{3.33333pt}{0ex}}{\Omega}^{2}\to (0,\infty )$ is a positive weight on ${\Omega}^{2}=\Omega \times \Omega $ with respect to ${\mu}^{2}=\mu \times \mu $ on ${\Omega}^{2}$ with

**Theorem 5.1.**Let X be a Banach space, let E be a metric space, $\sigma ,{\sigma}_{0},{\sigma}_{1}\in [1,\infty ]$ and ${c}_{J+},{c}_{J}>0$. Then one has:

**Corollary 5.2.**The proof of Theorem 5.1 also shows that for $[{\sigma}_{1},{\sigma}_{0}]\subset [1,\infty ]$, we have

**Definition 5.2.**Let X be a Banach space and $p,{p}_{0},{p}_{1}\in (0,\infty ]$, ${c}_{cl}>0$. We say that X is in Clarkson class $Cl({p}_{0},{p}_{1},{c}_{cl})$ if

**Lemma 5.2.**Let X be a Banach space and $p,{p}_{0},{p}_{1}\in [1,\infty ]$, ${c}_{cl}>0$. Then we have

**Remark 5.2.**Parts $\left(a\right)$ of Theorem 5.1 and Lemma 5.2 (and the majority of the results below) explain the importance of the sharpness of the Jacobi and Clarkson constants. In this article we apply the approach leading to precise constants for limited ranges of parameters. An alternative and somewhat less precise approach based on our counterparts of the Pythagorean theorem for Birkhoff-Fortet orthogonality is presented in [2].

#### 5.1.1. Constructive Approach for Particular Spaces

**Lemma 5.3.**For a Banach space X and a probability measure space $(\Omega ,\mu )$, one has

**Theorem 5.2**([45]). For ${p}_{0},{p}_{1}\in {[1,\infty ]}^{n}$, $\theta \in (0,1)$ and $1/{p}_{\theta}=(1-\theta )/{p}_{0}+\theta /{p}_{1}$, one has

**Theorem 5.3.**Let Y be a Banach space from the class $I{G}_{+}$ with $[{p}_{min}\left(Y\right),{p}_{max}\left(Y\right)]\in [1,\infty ]$ and finite or countable set $\Omega =I$. Let also X be either finitely represented (see Definition 7.1) in Y or a subspace, or a quotient of Y. Assume also that ${\left\{{x}_{i}\right\}}_{i\in I}\subset X$, ${\left\{{\alpha}_{i}\right\}}_{i\in I}\subset [0,1]$ with

**Remark 5.3**

**Theorem 5.4.**Let $Y\in \{{L}_{p}(\mathcal{M},\tau ),{S}_{p}\}$ with $p\in [1,\infty ]$, where $(\mathcal{M},\tau )$ is a von Neumann algebra with a normal semifinite faithful weight τ, and let $\Omega =I$ be finite or countable set. Let also X be either finitely represented (see Definition $7.1)$ in Y or a subspace, or a quotient of Y. Assume also that ${\left\{{x}_{i}\right\}}_{i\in I}\subset X$, ${\left\{{\alpha}_{i}\right\}}_{i\in I}\subset [0,1]$ with

**Theorem 5.5.**Let Y be a Banach space from the class $I{G}_{+}$ with $[{p}_{min}\left(X\right),{p}_{max}\left(X\right)]\in [1,\infty ]$. Let also X be either finitely represented (see Definition $7.1)$ in Y or a subspace, or a quotient of Y. Then one has:

**Remark 5.4.**

**Theorem 5.6.**Let $Y\in \{{L}_{p}(\mathcal{M},\tau ),{S}_{p}\}$ with $p\in [1,\infty ]$, where $(\mathcal{M},\tau )$ is a von Neumann algebra with a normal semifinite faithful weight τ. Let also X be either finitely represented (see Definition $7.1)$ in Y or a subspace, or a quotient of Y. Then one has:

#### 5.2. Quantitative Hahn-Banach Theorems

#### 5.2.1. Pichugov Classes

**Definition 5.3.**Let X be a Banach space, $\sigma \in [1,\infty ]$ and ${c}_{P+},{c}_{P}>0$. We say that the space X belongs to the adjoint Pichugov class ${P}_{+}\left(\sigma \right)={P}_{+}(\sigma ,{c}_{P+})$, or the Pichugov class $P\left(\sigma \right)=P(\sigma ,{c}_{P})$, if, for every combination ξ, η, ${\xi}^{\prime}$ and ${\eta}^{\prime}$ of independent X-valued stochastic variables, such that the variables in the pairs ξ and ${\xi}^{\prime}$ and η and ${\eta}^{\prime}$ are (independent and) identically distributed on a probability measure space $(\Omega ,\Xi ,p)$, one has, respectively, the estimates

**Remark 5.5.**

**Theorem 5.7.**Let X be a Banach space, $\sigma \in [1,\infty ]$ and ${c}_{P+},{c}_{P}>0$. Then one has:

**Theorem 5.8.**Let Y be a Banach space from the class $I{G}_{+}$ with $[{p}_{min}\left(Y\right),{p}_{max}\left(Y\right)]\in {[1,\infty ]}^{n}$. Let also X be either finitely represented (see Definition $7.1)$ in Y or a subspace, or a quotient of Y. Then we have:

**Theorem 5.9.**Let $Y\in \{{L}_{p}(\mathcal{M},\tau ),{S}_{p}\}$ with $p\in [1,\infty ]$, where $(\mathcal{M},\tau )$ is a von Neumann algebra with a normal semifinite faithful weight τ. Let also X be either finitely represented (see Definition $7.1)$ in Y or a subspace, or a quotient of Y. Then one has:

#### 5.2.2. Dol’nikov-Pichugov and Mutual Diameter Constants

**Definition 5.4.**Let X be a Banach space.

**Lemma 5.4.**Let X and Y be Banach spaces and $p\in [1,\infty ]$.

**Theorem 5.10**(Kirchberger). For $n\in \mathbb{N}$, let X be an n-dimensional normed space, and $E,F$ be its finite subsets. Then $\mathrm{co}\left(E\right)\cap \mathrm{co}\left(F\right)=\varnothing $ if, and only if, $\mathrm{co}(E\cap S)\cap \mathrm{co}(F\cap S)=\varnothing $ for every $S\subset E\cup F$ with $\left|S\right|\le n+2$.

**Theorem 5.11.**Let X be a Banach space and $r\in [1,\infty ]$. Assume also that A and B are bounded subsets of X with $d(A,B)=d>0$ and $X\in P(r,{c}_{P})\cap {P}_{+}(r,{c}_{P+})$. Then we have:

**Remark 5.6.**Pichugov [4] has also proved that, for $\sigma =r=1$ and an arbitrary Banach space X, one has

**Theorem 5.12**Let Y be a Banach space from the class $IG$ with $[{p}_{min}\left(Y\right),{p}_{max}\left(Y\right)]\in [1,\infty ]$. Let also X be either finitely represented (see Definition $7.1)$ in Y or a subspace, or a quotient of Y. Assume also that $r\in [1,min({p}_{min}\left(Y\right),{p}_{max}{\left(Y\right)}^{\prime})]$. Then we have:

## 6. Extension of Hölder-Lipschitz Mappings

**Definition 6.1**Assume that X is a metric space, Y is a Banach space, and $\alpha ,d>0$. Let ${H}^{\alpha}(X,Y)$ be the Banach space of all Y-valued continuous functions f defined on X with the finite norm:

#### 6.1. Isometric Extensions

**Definition 6.2**([9,10]). Assume that E is a set, and Y is a linear space. A function $\Phi :\phantom{\rule{3.33333pt}{0ex}}E\times E\times Y$ is called a K-function provided that $\Phi ({x}_{1},{x}_{2},y)$ is convex in y for every pair $({x}_{1},{x}_{2})\in E\times E$, and, for some $C>0$ and every $n\in \mathbb{N}$, finite sequence ${\left\{({x}_{i},{y}_{i})\right\}}_{i=1}^{n}\subset E\times Y$, $\alpha \in {S}_{n}$ and $x\in E$, one has

**Theorem 6.1**([9,10]). If, under the conditions of Definition $6.2$, the pairs ${\left\{({x}_{i},{y}_{i})\right\}}_{i=1}^{n}\subset E\times Y$ are such that

**Theorem 6.2.**For $2\in [p,q]\subset [1,\infty )$ and Banach spaces X and Y and a metric space $(E,d)$, let $X,E\in {J}^{a}(p,{c}_{J})$ and $Y\in {J}_{+}^{a}(q,{c}_{J+})$ with ${c}_{J}^{p}{c}_{J+}^{q}\le 1$. Then the convex $(1,\alpha )$-extension property for $\alpha \in (0,p/q]$ is possessed by the pairs $(X,Y)$ and $(E,Y)$.

**Remark 6.1.**

**Corollary 6.1.**If, under the conditions of Theorem $6.2$, E is an arbitrary metric space (i.e., $p=1$ and ${c}_{J}\left(E\right)=2$ according to Part $\left(b\right)$ of Remark $6.1)$, then the pair $(E,Y)$ possesses the convex $(1,\alpha )$-extension property for $\alpha \in (0,1/q]$.

**Theorem 6.3.**For $p\subset [1,\infty )$, $q\in (1,\infty )$ and Banach spaces X and Y, let ${l}_{p}$ and ${l}_{q}$ be finitely represented (see Definition $7.1)$ in X and Y correspondingly. Then we have

#### 6.2. Pairs of Concrete Spaces

**Theorem 6.4.**For $m,n\in \mathbb{N}$, ${s}_{0}\in {\mathbb{R}}^{n}$, ${s}_{1}\in {\mathbb{R}}^{m}$, ${q}_{0},{q}_{1},{q}_{2}\in (1,\infty )$ and ${p}_{0},{p}_{1},{p}_{2}\in {(1,\infty )}^{n}$, open subsets ${G}_{0}\subset {\mathbb{R}}^{n}$ and ${G}_{1}\subset {\mathbb{R}}^{m}$, let X be the ${l}_{2}$-sum of ${B}_{{p}_{0},{q}_{0}}^{s}{\left({\mathbb{R}}^{n}\right)}_{w}$ and ${B}_{{p}_{1},{q}_{1}}^{s}{\left({\mathbb{R}}^{m}\right)}_{w}$, and let also Y be the Bochner-Lebesgue space ${L}_{{p}_{2}}(E,{S}_{{q}_{2}})$ of the Schatten-von-Neumann-valued functions, Bochner-measurable on $E\subset {\mathbb{R}}^{n}$. Then we have

## 7. Miscellaneous Constants and Auxiliary Results

#### 7.1. Radon-Nikodým Property and Superreflexive and UMD Spaces

**Definition 7.1.**Let $X,Y$ be (quasi)Banach spaces and $\lambda \ge 1$. Then the Banach-Mazur distance ${d}_{BM}(X,Y)$ between them is equal to ∞ if they are not isomorphic and is, otherwise, defined by

**Definition 7.2.**A Banach space X is said to be superreflexive if it is reflexive and every Banach space Y finitely represented in X is reflexive too.

**Definition 7.3.**Let $p\in (1,\infty )$. A Banach space X is said to be a UMD space if there is a constant ${\beta}_{p}={\beta}_{p}\left(X\right)$ such that, for every X-valued martingale difference sequence ${\left\{{\xi}_{i}\right\}}_{i=1}^{\infty}$ on any probability space $(\Omega ,\sigma ,\nu )$ and every sequence ${\left\{{\epsilon}_{i}\right\}}_{i=1}^{\infty}$ of numbers 1 and $-1$, one has

#### 7.2. James, Jung, Self-Jung and Schäffer Constants

**Definition 7.4.**Let $A,B\subset X$ be a Banach space and its bounded subsets. A point $x\in B$ is a Chebyshev centre of A relative to B if $r(A,\{x\left\}\right)=r(A,B)$. The set of all such points is designated by ${C}_{X}(A,B)$. Setting $B=\overline{\mathrm{co}}\left(A\right)$, one obtains the self-Chebyshev centre ${C}_{X}(A,\overline{\mathrm{co}}\left(A\right))$ of A.