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This article is cited in 3 scientific papers (total in 3 papers) Continuous selections of set-valued mappings and approximation in asymmetric and semilinear spaces I. G. Tsar'kovab a Lomonosov Moscow State University, Faculty of Mechanics and Mathematics b Moscow Center for Fundamental and Applied Mathematics
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     § 1. IntroductionIn the present paper we consider linear spaces with asymmetric norm $\|\,{\cdot}\,|$, which are more general than classical normed linear spaces; we will also deal with asymmetric metric spaces and metric semilinear spaces, the latter being more general than metric linear spaces. An important example of such a space is the space of all closed bounded sets equipped with the Hausdorff metric. For numerous results on general properties of asymmetric spaces, and various problems of geometric approximation theory, see, for example, [1]–[16]. The purpose of the present paper is to put forward new theorems on continuous selection of set-valued mappings; as such mappings we consider, in particular, the (set-valued) operators of best or near-best approximation (in the latter setting, we speak about $\varepsilon$-selections). In the first part of the paper, we obtain an extension of the classical Michael continuous selection theorem for set-valued mappings with not necessarily convex values (Theorems 1 and 2). To this end, we will work with $\mathring{B}$-infinitely connected subsets of normed linear spaces. Such sets, which are more general than convex sets, will be considered as values of set-valued functions. Among numerous extensions of Michael’s theorem, we mention the survey [17] and the paper [18]. As a corollary, we will obtain a fixed-point theorem for lower semicontinuous mappings $\Psi\colon \mathcal{K}\to 2^\mathcal{K}$ with $\mathring{B}$-infinitely connected values defined on a $\mathring{B}$-infinitely connected compact set (Corollary 1). Next, we will study conditions under which any convex subset of an asymmetric normed linear space admits a continuous $\varepsilon$-selection for all $\varepsilon>0$ (Theorem 6). It is worth pointing out that, in essentially asymmetric spaces, this problem is quite challenging, in contrast to the classical (symmetric) normed spaces. Definition 1. Let $(X,\varrho)$ be a metric or a semimetric space (symmetric or asymmetric), $\varepsilon>0$, $M\subset X$. We say that $\varphi\colon X\to M$ is an additive (multiplicative) $\varepsilon$-selection if, for all $x\in X$, (respectively, $\varrho(x,\varphi(x))\leqslant (1+\varepsilon)\varrho(x,M)$), where $\varrho(x,M)=\inf_{y\in M}\varrho(x,y)$. We will also consider a classical example of a semilinear space, and show that in the space $\mathbf{L}_h(X)$, consisting of all convex non-empty bounded subsets of a reflexive space $X$, and equipped with the Hausdorff semimetric, any bounded set admits a Chebyshev centre (Theorem 9). § 2. Generalization of Michael’s theorem to set-valued mappings with infinitely connected valuesDefinition 2. We say that $\nu\colon X\times X\to \mathbb{R}_+$ is an asymmetric semimetric on $X$ if: 1) $\nu(x,x)= 0$ for all $x\in X$; 2) $\nu(x,z)\leqslant \nu(x,y)+ \nu(y,z)$ for all $x,y,z\in X$. In this case, we say that $\mathcal{X}=(X,\nu)$ is an asymmetric semimetric space. The function $\sigma(x,y):=\max\{\nu(x,y), \nu(y,x)\}$ is known as the symmetrization semimetric. A space $\mathcal{X}$ is said to be complete if it is complete with respect to the semimetric $\sigma$. Definition 3. A subset $A$ of a (symmetric or asymmetric) semimetric space $(X,\nu) $ is called infinitely connected if, for all $n\in \mathbb{N}$, any continuous mapping $\varphi\colon \partial B\to A$ of the boundary $ \partial B$ of the unit ball $B\subset \mathbb{R}^n$ has a continuous extension $\widetilde{\varphi}\colon B\to A$ to the whole of the unit ball $B$. A set $M\subset X$ is called $\mathring{B}$-infinitely connected ($B$-infinitely connected) if its intersection with any open (closed) ball is either infinitely infinitely connected or empty. A set $M\subset X $ is called $\mathring{B}$-contractible ($B$-contractible) if its intersection with any open (closed) ball is contractible or empty. Given a subset $M$ of an asymmetric semimetric space $(X,\nu)$, and $y\in X$, the distance to $M$ from $y$ (the right metric function) is defined by $\varrho(y,M)=\inf_{z\in M}\nu(y,z)$. The distance from a set $M$ to a point $y$ (the left metric function) is defined similarly: $\varrho^-(y,M)=\inf_{z\in M}\nu(z,y)$. The sets of all nearest points (left nearest points) in $M$ for $x$ are defined by, respectively,
$$
\begin{equation*}
P_Mx:= \{y\in M\mid \varrho(x,y)=\varrho(x,M), \qquad P _M^-x:=\{y\in M\mid \varrho(y,x)=\varrho^-(x,M)\}.
\end{equation*}
\notag
$$
The mapping $x\mapsto P_Mx $ (respectively, $x\mapsto P_M^-x $) is called the right (left) metric projection operator. We note the following simple properties of the distance function $\varrho(\,{\cdot}\,,M)$: 1. $\varrho(x,M)\leqslant \varrho(y,M)+\varrho(x,y)\ \forall \, x,y\in X$. Indeed, since $\varrho(x,M)\leqslant \varrho(x,z)\leqslant \varrho(x,y)+\varrho(y,z)$ for all $z\in M$, we have $\varrho(x,M)\leqslant \inf_{z\in M}(\varrho(x,y)+\varrho(y,z))= \varrho(y,M)+\varrho(x,y)$. 2. $|\varrho(x,M)- \varrho(y,M)|\leqslant \max\{\varrho(x,y),\varrho(y,x)\}=\sigma(x,y)\ \forall \, x,y\in X$. Definition 4. Let $(X,q)$ be a semimetric space. A mapping $\chi\colon X\to \overline{\mathbb{R}}$ is lower semicontinuous on $X$ if $\varliminf_{n\to\infty}\chi(x_n)\geqslant \chi(x)$ whenever $x\in X$ and $\{x_n\}\subset X$, $ q(x,x_n)\to 0$ as $n\to\infty$. Definition 5. Let $(X,q)$ and $(Y,\nu)$ be semimetric spaces, $M\subset Y$. A mapping $F\colon X\to 2^M$ is called lower stable if $F(x)\neq \varnothing$ for all $x\in X$, and if, for any $x_0\in X$ and any $\varepsilon>0$, there exists $\delta>0$ such that $\varrho(y,F(x))-\varrho(y,F(x_0))\leqslant\varepsilon$ for all $y\in Y$ and $x\in X$ with $q(x,x_0)\leqslant\delta$. Remark 1. Let $A\subset X$ be a closed subset of a semimetric space $(X,q)$, and let a mapping $F\colon A\to 2^M$, where $M\subset Y$ and $(A,q)$, $(Y,\nu) $ are semimetric spaces, be lower stable. Then the mapping $F$ defined by $F(x)=M$ for all $x\in X\setminus A$ is lower stable on the entire $X$. Given two non-empty subsets $A,B\subset Y$, the directed Hausdorff distance from $B$ to $A$ is defined by $d(A,B):=\sup_{y\in B}\varrho(y,A)$. The Hausdorff distance between $A$ and $B$ is defined by Note that $\sup_{y\in Y}(\varrho(y,F(x))-\varrho(y,F(x_0)))$ is the directed Hausdorff distance between $F(x)$ and $F(x_0)$. A compact-valued mapping $F$ is lower stable if and only if is lower semicontinuous. In what follows, we will consider seminormed spaces. A seminormed space is a linear space equipped with a (symmetric) seminorm $\|\,{\cdot}\,\|$; the difference of a seminorm from a norm is that the equality $\|x\|=0$ does not generally imply that $x=0$. Note that $\varrho(x,y):=\|y-x\|$ is a (symmetric) semimetric. An asymmetrically normed space (or an asymmetric space) $(X,\|\,{\cdot}\,|)$ is a linear space $X$ equipped with an asymmetric norm, which is defined by the following axioms: 1) $\|\alpha x|=\alpha\| x|$ for all $\alpha\geqslant 0$, $x\in X$; 2) $\| x+y|\leqslant \| x |+\| y| $ for all $x,y\in X$; 3) $\|x|\geqslant 0$ for all $x\in X$; 3a) $\|x|= 0\ \Leftrightarrow \ x=0$. Any asymmetric norm defines on $X$ an asymmetric metric $\nu(x,y):=\|y-x|$. In what follows, we will assume that $(X,\|\,{\cdot}\,\|)$, $(Y,\|\,{\cdot}\,\|) $ are (symmetric) seminormed linear spaces, and $M\subset Y$. We will also assume that a mapping $F\colon X\times[0,\vartheta]\to 2^M$ satisfies the following conditions: 1) $F(\,{\cdot}\,,t)$ is lower stable for all $t\in [0,\vartheta ]$; 2) for all $t_1<t_2$ and an arbitrary bounded set $\mathcal{A}\subset X$, there exists $\delta>0$ such that $O_\delta(F(x,t_1))\subset F(x,t_2)$ for all $x \in \mathcal{A}$; 3) the function $d(F(x,0),F(x,t))$ is continuous with respect to $x$ for each $t\in [0,\vartheta ]$, and tends to zero uniformly as $t\to 0+$ on any bounded set $\mathcal{A}\subset X$; 4) the set $F(x,t)$ is infinitely connected for all $x\in X$ and $t\in (0,\delta_0)$. We also define $\Psi_{\delta}(x,t) := \{u\in M\mid \varrho(u,F(x,t))<\delta\}=M\cap O_\delta(F(x,t))$, where $(x,t)\in X\times [0,\vartheta]$ and $\delta\in (0,\delta_0)$. Below, $\chi\colon X\to \overline{\mathbb{R}}$ is a given lower semicontinuous positive function. By replacing, if necessary, $\chi$ by $\min\{\chi(\,{\cdot}\,),\delta_0\}$ (for some $\delta_0>0)$ it can be assumed without loss of generality that $\chi(\,{\cdot}\,)\leqslant\delta_0$. In the proof of the following Lemmas 1–3, we use a modification of the technique in [16]. Lemma 1. Let $S\subset X$ be a non-degenerate simplex of dimension $n$, and $Q$ be the union of some proper faces of $S$. Next, let $\varphi\colon Q\to M$ be a continuous mapping such that $\varphi(x)\in F(x,t_0)$, for some $t_0\in (0,\vartheta)$, and Then there exists a continuous extension $\varphi\colon S\to M$ such that $\varphi(x)\in F(x,t)$ for some $t \in (t_0,\vartheta)$, and $\varrho(\varphi(x),F(x,0))\leqslant d(F(x,0),F(x,t))<\chi(x)$ for all $x\in S$. Proof. There exists a number $t>t_0$ such that Let $t_0<t'_0<t_1<t'_1<\dots<t_n<t'_n<t_{n+1}=t$. There exist $\delta_k>0$ such that for all $x \in S$, $k=1,\dots,n+1$. We set $\delta=(1/2)\min_{k=1,\dots,n+1}\delta_k$.
For each point $x\in S$, there exists a number $r=r(x)\in (0,\delta)$ such that, for any $y\in O_{2r}(x)$,
$$
\begin{equation*}
\sup_{u\in Y}\bigl(\varrho\bigl(u,F(y,t_k)\bigr)-\varrho\bigl(u,F(x,t_k)\bigr)\bigr)<\delta,\qquad k=1,\dots,n.
\end{equation*}
\notag
$$
Let $\{O_{r(x_j)}(x_j)\}_{j=1}^{N}$ be a finite subcovering of the open covering $\{O_{r(x)}(x)\}$ of $S$. We partition the simplex $S$ into a finite number of $n$-dimensional simplexes $\{S_i\}$ of diameter $<\delta$ requiring that each simplex $S_i$ be contained in at least one neighbourhood $O_{r(x_j)}(x_j)$. In addition, we assume that distinct simplexes of this partition intersect in at most one proper face of dimension $<n$.
$1^\circ$. For each zero-dimensional face (a vertex) $s$ of any simplex $S_i$, there exists an index $j=j_s$ for which the number $r(x_j)$ is maximal among those for which the face $s$ lies in the neighbourhood $O_{r(x_j)}(x_j)$. If $s\notin Q$, then with the point $s$ we associate some point $y := \varphi(s)\in F(x_j,t_0)\subset F(x_j,t'_0)$. We have
$$
\begin{equation*}
\varrho\bigl(\varphi(s),F(s,t'_0)\bigr)< \varrho\bigl(\varphi(s),F(x_j,t'_0)\bigr)+\delta=\delta,
\end{equation*}
\notag
$$
that is, $\varphi(s)\in \Psi_{\delta}(s,t'_0)$. If $s\in Q$, then $\varrho(\varphi(s),F(s,t'_0))=0< \delta$, that is, $\varphi(s)\in \Psi_{\delta}(s,t'_0)\subset F(s,t_1)$ for each zero-dimensional face $s$.
$2^\circ$. Assume that the mapping $\varphi$ extends continuously to the set $T_m$ defined as the union of the faces $\{\Delta_{ik}\}$ of dimension $m$ of simplexes in the family $\{S_i\}$. Note that $\varphi(x)\in F(x,t_{m+1})$ for all $x\in T_m$. Moreover, for each face $\Delta_{ik}$, there exists an index $j=j_{ik}=j_{\Delta_{ik}}$ such that $\Delta_{ik}\subset O_{r(x_j)}(x_j)$ and $\varphi(\Delta_{ik})\subset F(x_j,t'_{m})$. We take an arbitrary $(m+1)$-dimensional face $\Delta\not\subset Q$ of some simplex in the family $\{S_i\}$. The function $\varphi\colon T_m\to M$ is defined on its relative boundary $\partial\Delta$. With each face $P$ of the boundary $\partial\Delta$ of dimension $k\leqslant m$ we associate the greatest of the numbers $r(x_l)$, $(l=l_P)$, for which this face lies in the neighbourhood $O_{r(x_l)}(x_l)$. These neighbourhoods will also be associated with a face $P$. Let $j=j_{\Delta}$ be an index for which the number $r(x_j)$ is maximal among the numbers $r(x_k)$ for which the face $\Delta$ lies in $O_{r(x_k)}(x_k)$. For each face $\Delta_{ik}$ in the boundary $\partial\Delta$, the values of the function $\varphi$ on this face lie in $F(x_{j_{ik}},t'_m)\subset F(x_{j_{ik}},t_{m+1})$ $({j_{ik}}=j_{\Delta_{ik}})$. Taking into account that the neighbourhood $O_{r(x_{j_{ik}})}(x_{j_{ik}})$ has the maximal radius among those neighbourhoods that contain the face $\Delta_{ik}$, we get the inequality $r(x_{j_{ik}})\geqslant r(x_j)$. Since $x_j\in O_{2r(x_{j_{ik}})}(x_{j_{ik}})$, this implies that
$$
\begin{equation*}
\varrho\bigl(y,F(x_{j},t_{m+1})\bigr)-\varrho\bigl(y,F(x_{j_{ik}},t_{m+1})\bigr) <\delta.
\end{equation*}
\notag
$$
Therefore,
$$
\begin{equation*}
\varrho\bigl(\varphi(x),F(x_{j},t_{m+1})\bigr)\leqslant \varrho\bigl(\varphi(x),F(x_{j_{ik}},t_{m+1})\bigr)+\delta, \qquad x\in \Delta_{ik},
\end{equation*}
\notag
$$
and hence $\varphi(x)\in \Psi_{\delta}(x_j,t_{m+1})$, $x\in \Delta_{ik}$. As a result,
$$
\begin{equation*}
\varphi(\partial\Delta)\subset \Psi_{\delta}(x_j,t_{m+1})\subset F(x_j,t'_{m+1}).
\end{equation*}
\notag
$$
Since $F(x_j,t'_{m+1})$ is infinitely connected, there exists a continuous extension of $\varphi$ to $\Delta$ such that $\varphi(\Delta)\subset F(x_j,t'_{m+1})$. Since
$$
\begin{equation*}
\varrho\bigl(\varphi(x),F(x,t'_{m+1})\bigr)\leqslant \varrho\bigl(\varphi(x),F(x_j,t'_{m+1})\bigr)+\delta
\end{equation*}
\notag
$$
for all $x\in \Delta$, we have $\varphi(x)\in \Psi_{\delta}(x, t'_{m+1})\subset F(x,t_{m+2})$ for all $x\in \Delta$.
Since distinct $(m+ 1)$-dimensional faces intersect only in proper faces, the mapping $\varphi$ is well defined on the set $T_{m+1}$ that is the union of all faces of $(m+ 1)$-dimensional simplexes in the family $\{S_i\}$. Note that $\varphi \in C(T_{m+1})$. $3^\circ$. By induction, on $T_n = S$ we can construct a continuous mapping $\varphi\colon T_n\to M$ such that $\varphi(x)\in F(x,t)$ $(x\in S)$. This shows that, for all $y\in S $,
$$
\begin{equation*}
\varrho\bigl(\varphi(y),F(y,0)\bigr)\leqslant d\bigl(F(y,0),F(y,t)\bigr)<\chi(y),
\end{equation*}
\notag
$$
that is, $\varphi$ is the required mapping. Lemma 1 is proved. Remark 2. The proof of Lemma 1 shows that if $y\in S_i$ does not lie initially in $Q$, then $\varphi(y)\in F(x_j,t)$ for some index $j$ for which $y\in O_{r(x_j)}(x_j)$, and Hence, if the set $Q$ was empty at the beginning of the construction, then any point $y$ from any simplex $S_i$ has this property. Lemma 2. Let $(X,\|\,{\cdot}\,\|)$ and $(Y,\|\,{\cdot}\,\|) $ be seminormed linear spaces, $M\subset Y$, $\{x_\alpha\}_\alpha$ be a family of vectors in $X$, $\Sigma=\{S_\beta\}_\beta$ be a family of simplexes whose vertices form a finite subfamily of $\{x_\alpha\}_\alpha$ under the condition that any two distinct simplexes $S_\beta$ either have no common points, or intersect in some proper faces of them, or one of them is a proper face of the other. Next, let $T$ be the union of some simplexes in $\Sigma$ and let $F$ be the set-valued mapping defined above. Then there exists a mapping $\varphi\colon T\to M$ such that $\varrho(\varphi(x),F(x,0))<\chi(x)$ for all $x\in T$, the mapping $\varphi$ being continuous on each simplex in $T$. Proof. It can be assumed without loss of generality that all faces of simplexes $S_\beta$ also lie in the family $\Sigma$. For each zero-dimensional simplex $s$ in $\Sigma $ (that is, for some point in the family $\{x_\alpha\}_\alpha)$, we denote by $\varphi(s)$ some element $y\in M$ such that $\varrho(y,F(s,0))<\chi(s)$.
$1^\circ$. By Lemma 1, there exists a continuous extension of the mapping $\varphi$ to each one-dimensional simplex $S\in \Sigma$ such that $\varrho(\varphi(y),F(y,0))<\chi(y)$ for all $y\in S$. $2^\circ$. Assume that the mapping $\varphi\colon T_m\to M$ is constructed on the union $T_m$ of all $m$-dimensional simplexes in $\Sigma$ and that $\varphi$ is continuous on each such simplex. In addition, for any $(m+1)$-dimensional simplex $S\in \Sigma$, the continuous mapping $\varphi$ satisfies $\varrho(\varphi(y),F(y,0))<\chi(y)$ for all $y\in \partial S$ on the relative boundary $\partial S$ of $S$. By Lemma 1, there also exists a continuous extension $\varphi\colon S\to M$ such that $\varrho(\varphi(y),F(y,0))<\chi(y)$ for all $y\in S$. This mapping is well defined on the union of all $(m+1)$-dimensional simplexes in $\Sigma$, because distinct $(m+1)$-dimensional simplexes intersect only in proper faces (if the intersection is non-empty). This mapping is continuous on any such simplex. $3^\circ$. So, we have constructed a mapping $\varphi\colon T\to M$ such that
$$
\begin{equation*}
\varrho\bigl(\varphi(x),F(x,0)\bigr)<\chi(x)\quad \text{for all}\quad x\in T,
\end{equation*}
\notag
$$
and which is continuous on any simplex composing $T$. Lemma 2 is proved. Remark 3. Since any finite-dimensional space $X$ can be represented as a set $T$ from Lemma 2, there exists a function $\varphi\in C(X,M)$ such that $\varrho(\varphi(x),F(x,0))<\chi(x))$ for all $x\in X$. Remark 4. Let $\Sigma$ be a family of simplexes from Lemma 2. Since the proof of this lemma and the construction of the function $\varphi$ depend on Lemma 1, it follows by Remark 2 that, for each simplex $S_i$ from the family $\Sigma$ (after a refinement), we have $\varphi(y)\in F(x_j,t)$ for some index $j$ such that $y\in O_{r(x_j)}(x_j)$, and Lemma 3. Let $(X,\|\,{\cdot}\,\|)$ and $(Y,\|\,{\cdot}\,\|) $ be normed linear spaces, $M\subset Y$, and $F$ be a set-valued mapping from Lemma 1. Then there exists a mapping $\widehat{\varphi}\in C(X, M)$ such that $\varrho (\widehat{\varphi}(x),F(x,0))<\chi(x) $ for all $x\in X$. Proof. By Remark 3, it suffices to consider the case $\dim X=\infty$. There exists a continuous function $\widehat{\chi}\colon X\to \mathbb{R}$ such that $0<\widehat{\chi}(x)<\chi(x)$ for all $x\in X$.
For each point $x\in X $, there exists a largest number $\delta=\delta(x)\in (0,1]$ such that $\inf_{O_{3\delta}(x)}\widehat{\chi}\geqslant 4\delta$. Note that if $O_{\delta(y)}(y)\cap O_{\delta(z)}(z)\neq\varnothing$, and, for definiteness $\delta(y)\geqslant \delta(z)$, then $z\in O_{2\delta(y)}(y)$. Hence $\delta(z)\geqslant \delta(y)/3$, for otherwise we would have $O_{3\delta(y)}(y)\supset O_{\delta'}(z)\supset O_{3\delta(z)}(z)$ $(\delta'>3\delta(z))$, which would imply $\widehat{\chi}\geqslant 4\delta(y)\geqslant 12 \delta(z)$ on $O_{\delta'}(z)\supset O_{3\delta(z)}(z)$, contradicting the maximality of $\delta(z)$. Let $r=r(x)\in (0,\delta(x)/3)$ be such that $\sup_{u\in Y}(\varrho(u,F(y,0))-\varrho(u,F(x,t)))<\delta=\delta(x)$ for any $y\in O_{2r}(x)$. Note that if $y\in O_{2r(x)}(x)$ ($O_{2r(x)}(x)\subset O_{\delta(x)}(x)$), then $r(x)<\delta(x)/3\leqslant\delta (y)$. Let $\{U_\alpha\}_\alpha$ be an open locally finite subcovering of the open covering $\{O_{r(x)}(x)\}$ For each $\alpha$, there exists a point $y_\alpha$ such that $U_\alpha\subset O_{r(y_\alpha)}(y_\alpha)$. Let $\{g_\alpha\}_\alpha$ be a partition of unity corresponding to this covering, that is, $g_\alpha\equiv 0$ outside $ U_\alpha$; $\sum_\alpha g_\alpha\equiv 1$ on $X$, $g_\alpha\geqslant 0$ for all $\alpha$. We take a family of vectors $\{x_\alpha\}$ such that $x_\alpha\in U_\alpha$ for all $\alpha$ and any finite subfamily of vectors is linearly independent. Consider the continuous mapping $q(t)=\sum_\alpha x_\alpha g_\alpha(t)$. For each point $y\in X$, the index set $\mathcal{A}=\{\alpha\mid y\in \overline{U_\alpha} \}=\{\alpha_1,\dots,\alpha_N\}$ is finite, and there exists a neighbourhood $O(y)$ such that $\{\alpha\mid U_\alpha\cap O(y)\neq\varnothing\}=\mathcal{A} =\{\alpha_1,\dots,\alpha_N\}$. Let an index $i$ be such that $\max_{j=1,\dots,N}r(y_{\alpha_j})=r(y_{\alpha_i}) := r_y=r$. Then $U_{\alpha_j}\subset O_{2r}(y_{\alpha_i})$ for all $j=1,\dots,N$. Therefore, the simplex $S_y$ with vertices at the points $\{x_{\alpha_1},\dots,x_{\alpha_N}\}$ is contained in $O_{2r}(y_{\alpha_i})$. All distinct simplexes of the form $S_y$ $(y\in X)$ are either disjoint, or intersect in some proper faces of them, or one of them is a proper face of another one. By Lemma 2, there exists a mapping $\varphi\colon T\to M$ on the set $T=\bigcup_yS_y$ which is continuous on each $S_y$ and such that $\varrho (\varphi(u),F(u,0))<\widehat{\chi}(u) $ for all $u\in T$. For the continuous function $q(u)=\sum_\alpha x_\alpha g_\alpha(u)\in S_y\subset O_{2r}(y_{\alpha_i})$, we have, for all $u\in O(y)$,
$$
\begin{equation*}
\begin{gathered} \, \varphi(q(u))\in F(y_{\alpha_i},t)\quad\text{for some index }\alpha_i, \\ \varrho\bigl(\varphi(q(u)),F(q(u),t)\bigr)\leqslant \varrho\bigl(\varphi(q(u)),F(y_{\alpha_i},t)\bigr)+\delta \leqslant\delta, \\ \varrho\bigl(\varphi(q(u)),F(u,0)\bigr)\leqslant\varrho\bigl(\varphi(q(u)),F(u,t)\bigr)+\delta\leqslant \varrho\bigl(\varphi(q(u)),F(y_{\alpha_i},t)\bigr)+2\delta\leqslant 2\delta. \end{gathered}
\end{equation*}
\notag
$$
We set $\widehat{\varphi}(y)=\varphi(q(y))$. Then $\widehat{\varphi}\in C(X,M)$ and $\varrho(\widehat{\varphi}(y),F(y,0))\leqslant 2\delta(y)\,{<}\,\chi(y)$ for all $y\in X$. Lemma is proved. Remark 5. The conclusion of Lemma 3 remains true if in it the space $(X,\|\,{\cdot}\,\|)$ is replaced by a metric linear space, and then this space is replaced by an arbitrary metric space $(D,\nu)$ via an isometrical embedding $\tau\colon D\to X$ into the space of continuous functions $X:= C_b(D,\mathbb{R})$ with the metric $\upsilon(f,g):=\sup_D|f(\,{\cdot}\,)-g(\,{\cdot}\,)|$, where $\tau(x):=\nu(x,{\cdot}\,)\in C_b(D,\mathbb{R})$. Next, the mapping $F$ extends to the entire space $C_b(D,\mathbb{R})$ by setting $F(x,t):=M$ for all points $x\notin \tau(X)$ and $t\in \mathbb{R}$. So, in Lemma 3, we can assume that $X$ is an arbitrary metric space. In the proof of the theorem, we will use Corollary 4 in [16] to the effect that any closed (open) $r$-neighbourhood ($r>0$) of a $\mathring{B} $-infinitely connected closed subset of a Banach space is $\mathring{B}$-infinitely connected, and, therefore, the intersection of this neighbourhood with any open ball is infinitely connected. In the next theorem, we also set $M=Y$. Note that in Theorem 1 the case of a seminormed space $(Y,\|\,{\cdot}\,\|)$ can be reduced to that of a normed space by taking the quotient of $Y$ with respect to the subspace $L:=\{y\in Y\mid\|y\|=0\}$ and equipping this quotient space with the quotient norm. Theorem 1. Let $(X,\upsilon)$ be a metric space, $(Y,\|\,{\cdot}\,\|)$ be a complete seminormed linear space, and let a set-valued mapping $\Phi\colon X\to 2^Y$ be lower stable and have closed $\mathring{B} $-infinitely connected values in $ (Y,\|\,{\cdot}\,\|) $. Then there exists a mapping $\varphi\in C(X, Y)$ such that $ \varphi(x)\in \Phi(x)$ for all $x\in X$. Proof. 1. By Lemma 3 and the remark preceding the theorem, there exists $\psi_1\in C(X, Y)$ such that $\varrho(\psi_1(x),\Phi(x))<1/2$ ($x\in X$).
2. Assume that we have already constructed a mapping $\psi_n\in C(X, Y)$ such that $ \varrho(\psi_n(x),\Phi(x))<1/2^n$. We set $\chi(x)=2^{-n-1}-(1/2)\varrho(\psi_n(x),\Phi(x))$ and $\delta(x)=(1/2)\varrho(\psi_n(x),\Phi(x))$. Consider the set-valued mappings $H(x,t)=O_{\delta(x)+t}(\Phi(x))$ and $G(x,t)=O_{2^{-n-1}+t}(\psi_n(x))$ ($x\in X$). For all $t\geqslant 0$, the sets $F(x,t)=H (x,t)\cap G (x,t)$ are non-empty and infinitely connected. In addition, it is easily checked that the mapping $F$ satisfies the conditions formulated before Lemma 1, and, therefore, for it the conclusions Lemma 3 and Remark 5 are true. Hence there exists a mapping $\psi_{n+1}\in C(X, Y)$ such that $ \varrho(\psi_{n+1}(x),F(x,0))<\chi(x)$ ($x\in X$). It follows that $\varrho(\psi_{n+1}(x),\Phi(x))<\delta(x)+\chi(x)=2^{-n-1}$ and $\|\psi_{n+1}(x)-\psi_n(x)\|<2^{-n}$. 3. The sequence $\{\psi_n(x)\}$ is a Cauchy sequence uniformly on $(Y,\|\,{\cdot}\,\|)$, and hence, converges to some continuous mapping $\varphi\colon X\to Y$. In addition, $\varrho(\psi_{n+1}(x),\Phi(x))<2^{-n-1}\to 0$ as $n\to\infty$, and, therefore, $\varrho(\varphi(x),\Phi(x))=0$, that is, $\varphi(x)\in \Phi(x)$ for all $x\in X$. Theorem 1 is proved. As in the above theorem, in the following result, the case of a seminormed space $ (X,\|\,{\cdot}\,\|)$ can be reduced to that of a normed space by taking the quotient over the space of elements with zero seminorm. Corollary 1. Let $ (X,\|\,{\cdot}\,\|) $ be a seminormed linear space, let $\mathcal{K}\subset X$ be a compact $\mathring{B}$-infinitely connected set, and let $\Psi\colon \mathcal{K}\to 2^\mathcal{K}$ be a lower semicontinuous mapping with $\mathring{B}$-infinitely connected compact values. Then there exists a fixed point $x_0\in \mathcal{K}$ such that $ x_0\in \Psi(x_0)$. Proof. Without loss of generality it can be assumed that $X$ is a complete space. Since $\mathcal{K}$ is a $\mathring{B}$-infinitely connected compact set, it admits a continuous multiplicative $\varepsilon$-selection $\psi\colon X\to \mathcal{K}$ (for some $\varepsilon>0$) which is a retraction of the entire space $X$ onto $\mathcal{K}$ (see Corollary 2 in [16]). Hence the mapping $\Phi=\Psi\circ \psi$ is lower stable. As already pointed out, any lower semicontinuous mapping with compact values is lower stable. Applying the previous result, we construct a continuous mapping $\varphi\colon X\to X$ such that $\varphi(x)\in \Phi(x)$ ($x\in X$).
By Schauder’s fixed point theorem, there exists a fixed point $x_0\in \mathcal{K}$ for the restriction of the continuous mapping $\varphi$ to the convex hull of the compact set $\mathcal{K}$. Hence $x_0=\varphi(x_0)\in \Phi(x_0)=\Psi(x_0)$, proving the corollary. There have been several attempts to obtain selection theorems for spaces more general than metrizable spaces and to find a satisfactory combination of metrical and convex type conditions. We note in this regard the studies of Michael [19], Beer [20], and the survey paper by Repovš and Semenov [17]. § 3. Continuous $\varepsilon$-selections in asymmetric metric spacesDefinition 6. A set $X$ will be called a semilinear space (or a cone) over $\mathbb{R}$ if the operation of addition of vectors and multiplication by non-negative scalars are defined on $X$ and if, for all $x,y,z\in X$ and $\alpha,\beta\in \mathbb{R}_+$: 1) $x+y=y+x$; 2) $x+(y+z)=(x+y)+z$; 3) there exists a unique zero element $\theta\in X$ such that $x+\theta=x$; 4) $\alpha(x+y)=\alpha x+\alpha y$; 5) $(\alpha+\beta)x=\alpha x+\beta y$; 6) $\alpha(\beta x)=(\alpha\beta)x$; 7) $0\cdot x=\theta$; $1\cdot x=x$. When no confusion will ensue, we denote the zero element $\theta$ by $0$. Definition 7. A pair $\mathcal{X}=(X,\varrho)$, where $X$ is a semilinear space, will be called an asymmetric semimetric semilinear space if $\varrho$ is an asymmetric semimetric on $X$, and if the operations of addition of elements and multiplication by non-negative scalars are continuous; that is, if $\{\alpha_n\},\{\beta_n\}\subset \mathbb{R}_+$ and $\{x_n\},\{y_n\}\subset X$ converge, respectively, to $\alpha,\beta\in \mathbb{R}_+$ and $x,y\in X$, then the sequence $\sigma(\alpha_n x_n+\beta_n y_n,\alpha x +\beta y)$ converges to $0$, where $\sigma$ is the symmetrization semimetric. A set $M\subset X$ is called convex if $[a,b]\subset M$ for all $a,b\in M$. An asymmetric semimetric $\varrho$ is called generalized convex (respectively, convex) on a convex set $M\subset X$ if $\varrho(z,\alpha x+(1-\alpha)y)\leqslant \max\{\varrho(z,x),\varrho(z,y)\}$ (respectively, $\varrho(z,\alpha x+(1-\alpha)y)\leqslant \alpha\varrho(z,x)+(1-\alpha)\varrho(z,y)\bigr)$ for all $\alpha\in[0,1]$, $z\in X$ and $x,y \in M$. In particular, if $\varrho$ is convex, then it is generalized convex. Remark 6. If an asymmetric semimetric is generalized convex on a $M$, then, for any interval interval $[x,y]\subset M$, Remark 7. If an asymmetric semimetric is generalized convex on a convex set $M$, then $M$ has contractible intersections with open balls, and, therefore, $M$ is $\mathring{B}_\varrho$-infinitely connected (that is, the intersection of $M$ with any open ball is infinitely connected). Hence by Theorem 5 in [12], the convex set $M$ admits a continuous $\varepsilon$-selection for any $\varepsilon>0$. The next result is Corollary 6 in [12]. Theorem A. Let $(X,\varrho)$ be a semimetric semilinear space with generalized convex semimetric $\varrho$ satisfying $\varrho(x,y)=\varrho(y,x)$ and $\varrho(x,(1-\alpha)x+\alpha y)= \alpha \varrho(x,y) $ for all $x,y\in X$ and $\alpha\in [0,1]$ (in particular, this condition is met for the space $\mathbf{L}_h$; see § 4 below). Also let $M $ be either $\mathring{B}_\varrho$-infinitely connected in $X$, or admit a continuous additive $\varepsilon$-selection for any $\varepsilon>0$. Then any $r$-neighbourhood $(r>0)$ of $M$ is $\mathring{B}_\varrho$-infinitely connected. Arguing precisely as in the proof of Theorem 1 and invoking Theorem A, we arrive at the following result, which extends Theorem 1 and generalizes the classical variant of Michael’s theorem. Theorem 2. Let $(X,\upsilon)$ be a metric space, $(Y,\varrho)$ be a complete semimetric semilinear space with generalized convex semimetric $\varrho$ satisfying $\varrho(x,y)=\varrho(y,x)$ and $\varrho(x,(1-\alpha)x+\alpha y)= \alpha \varrho(x,y) $ for all $x,y\in X$ and $\alpha\in [0,1]$. Next, let a set-valued mapping $\Phi\colon X\to 2^Y$ be lower stable and have closed $\mathring{B}$-infinitely connected values in $(Y,\varrho)$. Then there exists a mapping $\varphi\in C(X, Y)$ such that $ \varphi(x)\in \Phi(x)$ for all $x\in X$. Remark 8. In Remark 1 of [11] it was noted that in an asymmetric normed linear space $(Y,\|\,{\cdot}\,|)$ in which the asymmetric norm is equivalent to the symmetrization norm $\|\,{\cdot}\,\|_{\mathrm{sym}}$ (see the definition below), any (closed) $r$-neighbourhood of a $\mathring{B}$-infinitely connected set $M\subset Y$ is $\mathring{B}$-infinitely connected in $Y$. Using this fact, one can easily carry over Theorem 2 to the case where $Y$ is a complete asymmetric normed space in which the asymmetric norm is equivalent to the symmetrization norm. Corollary 2. Let either $(Y,\|\,{\cdot}\,|)$ be a complete asymmetric linear space in which the asymmetric norm is equivalent to the symmetrization norm or $(Y,\varrho)$ be a complete semimetric semilinear space in which the semimetric $\varrho$ is generalized convex and satisfies $\varrho(x,y)=\varrho(y,x)$ and $\varrho(x,(1-\alpha)x+\alpha y)= \alpha \varrho(x,y)$ for all $x,y\in X$ and $\alpha\in [0,1]$. Next, let $M\subset Y$ and the metric projection $P_M$ (respectively, $P_M^-)$ have closed $\mathring{B}$-infinitely connected values in $Y$. Then there exists a mapping $\varphi\in C(X, M)$ such that $ \varphi(x)\in P_M(x)$ (respectively, $ \varphi(x)\in P_M^-(x)$) for all $x\in X$. The next result is proved similarly. Corollary 3. Let either $(Y,\|\,{\cdot}\,|)$ be a complete asymmetric linear space in which the asymmetric norm is equivalent to the symmetrization norm or $(Y,\varrho)$ be a complete semimetric semilinear space in which the semimetric $\varrho$ is generalized convex and satisfies $\varrho(x,y)=\varrho(y,x)$ and $\varrho(x,(1-\alpha)x+\alpha y)= \alpha \varrho(x,y)$ for all $x,y\in X$ and $\alpha\in [0,1]$. Next, let $M\subset Y$ admit a set-valued selection $\Psi\colon X\to 2^M$ of the metric projection $P_M$ (respectively, $P_M^-)$ and $\Psi$ be lower stable and have closed $\mathring{B}$-infinitely connected values in $Y$. Then there exists a mapping $\varphi\in C(X, M)$ such that $ \varphi(x)\in \Psi(x)\subset P_M(x)$ (respectively, $\varphi(x)\in \Psi(x)\subset P_M^-(x)$) for all $x\in X$. Remark 9. Under the conditions of Corollary 2 (or Corollary 3), there exists a continuous selection of the metric projection operator, and hence, if a “closed” right ball $B(x,r):=\{y\in Y\mid \varrho(x,y)\leqslant r\}$ or a left ball $B^-(x,r):=\{y\in Y\mid \varrho(y,x)\leqslant r\}$ has non-empty intersection with $M$, then the corresponding continuous selection $\varphi$ maps the cone $K:=\{z\in [x,y]\mid y\in M\cap B(x,r)\}\subset M\cap B(x,r)$ (or, respectively, the cone $K^-:=\{z\in [x,y]\mid y\in M\cap B^-(x,r)\}\subset M\cap B^-(x,r)$) into itself and is identical on $M\cap B(x,r)$ (respectively, $M\cap B^-(x,r)$). As a result, in both these cases, the sets $ P_M(x)$ ($P_M^-(x)$) and $M\cap B(x,r)$ ($M\cap B^-(x,r)$) are retracts of the corresponding cone, and, therefore, are contractible in itself. Hence, the intersection of $M$ with any closed right (left) ball is contractible. Consider now asymmetric seminormed linear spaces. In the remaining part of this section, we will consider classes of linear spaces equipped with asymmetric (semi)norm $\|\,{\cdot}\,|$. An asymmetric norm on a linear space $X$ is defined by the following axioms: 1) $\|\alpha x|=\alpha\| x|$ for all $\alpha\geqslant 0$, $x\in X$; 2) $\| x+y|\leqslant \| x |+\| y|$ for all $x,y\in X$; 3) $\|x|\geqslant 0$ for all $x\in X$; 3a) $\|x|= 0 \ \Leftrightarrow \ x=0$. Any asymmetric norm is generated by the Minkowski functional of some (asymmetric, in general) body containing in its kernel the origin, which not necessarily lies in its centre of symmetry. In general, a space with an asymmetric norm satisfies only the $T_1$ separation axiom (that is, for all $a, b \in X$ one can find neighbourhoods $O(a)$, $O(b)$ such that $a\notin O(b)$, $b\notin O(a)$), and can fail to be Hausdorff. We will also consider asymmetric seminorms $\|\,{\cdot}\,|$, which are defined by axioms 1)–3), while axiom 3a) is replaced by the condition $(\|x|= 0=\|{-}x|)\, \Rightarrow\, x=0$. Spaces with asymmetric seminorm will be called asymmetric seminormed spaces. An asymmetric seminormed space $(X,\|\,{\cdot}\,|)$ satisfies only the $T_0$ separation axiom. The case of seminormed spaces will always be explicitly mentioned; otherwise a space will be assumed to be equipped with an asymmetric norm. In parallel with asymmetric (semi) norms $\|\,{\cdot}\,|$, it is often convenient to consider the symmetrization norm defined by
$$
\begin{equation*}
\|x\|=\|x\|_{\mathrm{sym}}:=\max\{\|x|,\|{-}x|\},\qquad x\in X.
\end{equation*}
\notag
$$
Given a set $M\subset X$, we will write $M^{\mathrm{sym}}$ if $M$ is considered in the topology generated by the symmetrization norm. We will also consider the symmetric seminorm $\|\,{\cdot}\,\|_{\Sigma}$ defined by According to [21], the seminorm $\|x\|_\Sigma$ is the greatest of the symmetric seminorms satisfying $\|\,{\cdot}\,\|_{\Sigma}\leqslant\|\,{\cdot}\,|,\|{-}\,{\cdot}\,|$. Given a linear asymmetric (semi)normed space, $B(x,r)= \{y\in X\mid \|y-x|\leqslant r\}$ and $\mathring{B}(x,r) = \{y\in X\mid \|y- x|< r\}$ are, respectively, the “closed” and open balls of radius $r$ with centre $x$. We also define the left balls $B^-(x,r)=\{y\in X\mid \|x- y|\leqslant r\}$ and $\mathring{B}^-(x,r) = \{y\in X\mid \|x- y|< r\}$. The (right) topology of $X$ is defined by the subbase generated by the open balls $\mathring{B}(x,r)$; the left topology is generated by the subbase of left open balls $\mathring{B}^-(x,r)$. Definition 8. Let $\varnothing \ne M\subset X$. We say that $x\in X\setminus M$ is a solar point for $M$ if there exists a point $y\in P_Mx\ne \varnothing$ (called a luminosity point) such that $y\in P_M((1-\lambda)y+\lambda x)$ for all $\lambda\geqslant 0$ (geometrically, this means that there is a “solar” ray emanating from $y$ and passing through $x$ such that $y$ is a nearest point in $M$ for any point on the ray). We say that $x\in X\setminus M$ is a strict solar point if $P_Mx\ne\varnothing$ and each $y\in P_Mx$ is a luminosity point for $x$. A set $M$ is a sun (a strict sun) if any point from $ X\setminus M$ is a solar (strict solar) point for $M$. The the following result can be found in [9]. Theorem B. Let $(X,\|\,{\cdot}\,|)$ be a finite-dimensional asymmetric space, $M\subset X$, and let the metric projection $P_M$ onto $M$ be lower semicontinuous. Then there exists a continuous mapping $\varphi\colon X\to M$ such that $\varphi(x)\in P_Mx$ for all $x\in X$. In Theorem 5 of [14] it was shown that, in any finite-dimensional asymmetric polyhedral space, a set with lower semicontinuous metric projection is strict sun. Hence from Theorem B we have following result. Theorem 3. Let $(X,\|\,{\cdot}\,|)$ be a finite-dimensional asymmetric space and let $M\subset X$ be a set with lower semicontinuous metric projection. Then there exists a continuous mapping $\varphi\colon X\to M$ such that $\varphi(x)\in P_Mx$ for all $x\in X$. Moreover, $M$ is $B$-contractible, and if, in addition, $X$ is polyhedral, then $M$ is a strict sun. Definition 9. A path $p\colon [0,1]\to X$ in an asymmetric linear space $X$ is said to be monotone if $x^*(p(\tau))$ is a monotone function for any norm-one extreme functional $x^*$. A set $M\subset X$ is called monotone path-connected if any two points from $M$ can be connected by a monotone path whose trace lies in $M$. A subset $M$ of a normed linear space $X$ is called stably monotone path-connected if there exists a continuous mapping $p\colon M\times M\times[0,1]\to M$ such that, for all $x,y\in M$, $p(x,y,{\cdot}\,)$ is a monotone path connecting $x$ and $y$. For normed spaces, monotone path-connectedness was introduced by Alimov [22]. It is known that any closed (open) ball is monotone path-connected and that the intersection of any closed (open) ball with any monotone path-connected set is monotone path-connected. Stably monotone path-connected sets (see [12], [14]) feature the same properties, and, in addition, are $B$- and $\mathring{B} $-contractible, and, therefore, $\mathring{B}$-infinitely connected. Any compact monotone path-connected set is $\mathring{B}$-infinitely connected. Now from Theorem 3 and Corollary 3 we have Theorem 4. Let $(Y,\|\,{\cdot}\,|)$ be a complete asymmetric linear space in which the asymmetric norm is equivalent to the symmetrization norm $\|\,{\cdot}\,\|_{\mathrm{sym}}$ and let $M\subset Y$ be either a stably monotone path-connected set or an approximatively compact (see Definition 11 below) monotone path-connected set. Then the following conditions are equivalent: 1) the set $M$ admits a lower stable (respectively, lower semicontinuous) set-valued selection $\Psi\colon X\to 2^M$ of the metric projection $P_M$ (respectively, $P_M^-$); 2) the set $M$ admits a continuous selection $\varphi\colon X\to M$ of the metric projection $P_M$ (respectively, $P_M^-$). Example. In the space of all continuous functions on a compact set $Q$, consider the asymmetric norm $\|f|_{\psi_+,\psi_-}:=\max_{x\in Q}\{f_+/\psi_+,f_-/\psi_-\}$, where $f_+=\max\{f,0\}$ and $f_-=\max\{-f,0\}$ for $f\in C(Q,\mathbb{R})$, and $\psi_+$ and $\psi_-$ are fixed functions satisfying $0<c<\psi_+,\psi_-<C$ for some positive constants $c$ and $C$. The asymmetric ball $B(0,R)$ consists of all functions $f$ lying between $R\psi_+$ and $-R\psi_-$, that is, $-R\psi_-(x) \leqslant f(x)\leqslant R\psi_+(x)$ for all $x\in Q$. Hence the ball $B(g,R)$ consists of all functions $f$ such that $f-g$ lies between $R\psi_+$ and $-R\psi_-$. Let $C_{\psi_+,\psi_-}(Q)$ be the space of continuous functions on a compact set $Q$ with asymmetric norm $\|\,{\cdot}\,|_{\psi_+,\psi_-}$. This norm is equivalent to the symmetrization norm, which, in turn, is equivalent to the classical uniform norm on $C(Q)$. A path $p(\,{\cdot}\,)\colon [0,1]\to C_{\psi_+,\psi_-}(Q) $ will be said to be monotone if $(p(\tau))[x]$ is a monotone function of $\tau$ for all $x\in Q$; a set $M\subset C_{\psi_+,\psi_-}(Q) $ will be said to be monotone path-connected if any two points $M$ can be connected by a monotone path whose trace lies in $M$. In this setting, Theorem 4 can be applied to the space $Y=C_{\psi_+,\psi_-}(Q) $ and to its approximatively compact subset $M$. In $C(Q)$, each boundedly weakly compact sun is monotone path-connected (see Theorem 4 in [23]), and hence from the above Theorem 4 we get following result. Theorem 5. Let $M$ be a boundedly weakly compact approximatively compact sun in $C(Q)$. Then the following conditions are equivalent: 1) $M$ admits a set-valued lower semicontinuous selection $\Psi\colon X\to 2^M$ of the metric projection operator $P_M$; 2) $M$ admits a continuous selection $\varphi\colon X\to M$ of the metric projection $P_M$. The following result is immediate from Corollary 1 in [12]. Theorem C. Let $(X,\|\,{\cdot}\,|) $ be an asymmetric seminormed linear space, $M\subset X$ be a $\mathring{B}$-infinitely connected (in particular, convex) set. Then, for any lower semicontinuous (with respect to the symmetrization norm) function $\psi\colon X\to\overline{\mathbb{R}}$ such that $ \varrho(x,M)<\psi(x)$ ($x\in X$), there exists a mapping $\varphi\in C(X^{\mathrm{sym}},M)$ such that $ \|\varphi(x)-x|<\psi(x)$ ($x\in X$). Similarly, for a non-empty open set $D\subset X$ and a lower semicontinuous (with respect to the symmetrization norm) function $\psi\colon D\to\overline{\mathbb{R}}$ such that $\varrho(x,M)<\psi(x)$ ($x\in D$), there exists a mapping $\varphi\in C(D^{\mathrm{sym}},M)$ such that $ \|\varphi(x)-x|<\psi(x)$ ($x\in D$). Theorem 6. Let $(X,\|\,{\cdot}\,|) $ be an asymmetric seminormed linear space, let $M\subset X$ be a convex set, and let $K\subset X$ be a non-empty Hausdorff compact set on which the metric function $\varrho(\,{\cdot}\,,M)$ is continuous. Then, for any $\varepsilon>0$, there exists a continuous function $\varphi\in C(K,M)$ such that Proof. Recall that metric function $\varrho(\,{\cdot}\,,M)$ (and, therefore, the function $\psi(\,{\cdot}\,)=\varrho(\,{\cdot}\,,M)+\varepsilon/2$) is lower semicontinuous on $X$ (see [7], [8]) and continuous on any finite-dimensional subspace. Given any $\varepsilon>0$, for each point $x$ of the compact set $K$, consider a $\delta_x$-neighbourhood $O_{\delta_x}(x)$, where $\delta_x\in (0,\varepsilon/4)$, of this point such that $\varrho(y,M)<\varrho(x,M)+\varepsilon/4$ for all $y\in O_{\delta_x}(x)$ (this is possible since the metric function is continuous). The open covering of $K$ by the neighbourhoods $\{O_{\delta_x}(x)\}_{x\in K}$ contains a finite subcovering. The centres of the neighbourhoods of this subcovering form a finite $\varepsilon/4$-net for $K$. Let $T=T_\varepsilon$ be the convex hull of this finite $\varepsilon/4$-net for $K$. Note that the restriction of the right (or left) topology to $T$ is equivalent to the topology $\tau$ generated by the symmetrization norm. Let We claim that $\Phi$ is right-left lower semicontinuous (this means that the pre-image is equipped with the right topology, and the image, with the topology $\tau$). Indeed, this claim follows from the following fact: if $\|z_n-z|\to 0$ as $n\to\infty$ and $y\in \Phi(z)$, then there exists $\delta\in (0,\varepsilon/4)$ such that $\|z-y|<\delta$, and $\|z_n-y|\leqslant \|z_n-z|+\|z-y|< \|z_n-z|+\delta<\varepsilon/4$ for all $n $ exceeding some $n_0$. Therefore, $y\in \Phi(z_n)$ for all $n>n_0$.
By Theorems 1 and 2, there exists a continuous mapping $g\colon K\to T$ such that $g(x)\in \Phi(x)$ (that is, $\|g(x)-x|<\varepsilon/4$) for $x\in K)$. By Theorem C, for the finite-dimensional subspace $T^0$ spanned by $T$, there exists a continuous function $\vartheta\colon T^0\to M$ such that $\|\vartheta(x)-x|<\varrho(x,M)+\varepsilon/2$ ($x\in T^0$). Hence $\varphi=\vartheta\circ g\colon K\to M$ is a continuous mapping, and $\|\varphi(x)-x|=\|\vartheta\circ g(x)-x|\leqslant \|\vartheta\circ g(x)- g(x)|+\|g(x)-x|<\varrho(g(x),M)+\varepsilon/2+\varepsilon/4<\varrho(x,M)+\varepsilon$. Theorem 6 is proved. Similarly, considering the left metric function (or, alternatively, multiplying all sets and vectors by $-1$, or equipping $X$ with the asymmetric seminorm $\|\,{\cdot}\,|^-:=\|-\,{\cdot}\,|$), we arrive at the following result. Theorem 7. Let $(X,\|\,{\cdot}\,|) $ be an asymmetric seminormed linear space, let $M\subset X$ be a convex set, and let $K\subset X$ be a non-empty Hausdorff compact set (in the left topology) on which the metric function $\varrho^-(\,{\cdot}\,,M)$ is continuous in the left topology. Then, for any $\varepsilon>0$, there exists a left-left continuous function (both the image and the pre-image are equipped with the left topology) $\varphi\colon K\to M$ such that Remark 10. For a paracompact asymmetric normed space $X$, Theorems 6 and 7 are also true if the compact set $K$ is replaced by the paracompact set $X$ (with respect to the right or left topology). In the proof of this result, we may avoid using Theorem C by constructing an appropriate locally finite covering and a corresponding partition of unity. Since any separable asymmetric normed space $X$ with closed unit ball is a Lindelöf space, $X$ is paracompact. Therefore, in this case, the set $K$ from Theorems 6 and 7 can be replaced by the entire space $X$. Definition 10. Let $X=(X,\|\,{\cdot}\,|)$ be a space with asymmetric seminorm. A set $M\subset X$ is a left- (right)-approximatively) stable at a point $x\in X$ if, for any $\{x_n\}\subset X$, any $\{y_n\}\subset M$, and any null sequence $\{\varepsilon_n\}\subset \mathbb{R}_+$ such that $\|x_n-x|\to 0$ as $n\to\infty$, the condition implies that there exists a subsequence $\{y_{n_k}\}$ such that $\|y-y_{n_k}|\to 0$ (respectively, $\|y_{n_k}-y|\to 0$) as $k\to\infty$ for some point $y\in M$. A set $M\subset X$ which is left- (right-)approximatively stable at each point $x\in X$ is said to be left- (right-)approximatively stable. Definition 11. Let $ M$ be a non-empty subset of an asymmetric seminormed space $(X,\|\,{\cdot}\,|)$. We say that $x\in X$ is a point of left (right) approximative compactness for $M$ (written $x \in \mathrm{AC}^-(M)$ ($x \in \mathrm{AC}(M)$) if, for any minimizing sequence $\{y_n\}\subset M$ (that is, $\|x-y_n|\to \varrho^-(x,M)$ (respectively, $\|y_n- x|\to \varrho(x,M)$ as $n\to\infty$)), there exists a subsequence $\{y_{n_k}\}$ such that $\|y-y_{n_k}|\to 0$ ($\|y_{n_k}-y|\to 0$) as $k\to\infty$ for some point $y \in M$. If $\mathrm{AC}^-(M)=X$ ($\mathrm{AC}(M)=X$), we say that $M$ is left- (right-)approximatively compact. The following theorem was proved in [24]. Theorem D. Let $X=(X,\|\,{\cdot}\,|)$ be an asymmetric seminormed space in which the unit ball $B(0,1)$ is closed and let $M\subset X$ be left-approximatively stable at $x\in X$. Then $x\in X$ is a point of left-approximative compactness for $M$, and the function $\varrho^-(\,{\cdot}\,,M)$ is continuous at $x$. The next result is a consequence of Remark 10 and Theorems 7 and D. Theorem 8. Let $X=(X,\|\,{\cdot}\,|)$ be an asymmetric seminormed space in which the unit ball $B(0,1)$ is closed and let $M\subset X$ be a left-approximatively stable convex set. Then, for any $\varepsilon>0$, there exists a continuous function $\varphi\in C(X,M)$ such that § 4. Existence of a Chebyshev centre in a space with Hausdorff semimetricDefinition 12. Let $\mathcal{X}=(X,\|\,{\cdot}\,\|)$ be a seminormed linear space. Consider the family $\mathcal{M}=\mathcal{M}(X)$ of all bounded subsets of $X$. Given two sets $M_1,M_2\in \mathcal{M}$, the linear combination $\alpha M_1+\beta M_2$ is defined by $\{z=\alpha x_1+\beta x_2\mid x_j\in M_j,\ j=1,2\}$. On the set $\mathcal{M}$, we consider the symmetric semimetric $h(\,{\cdot}\,,{\cdot}\,)$ defined as the Hausdorff distance; we also consider the asymmetric semimetric $d(\,{\cdot}\,,{\cdot}\,)$ defined as the directed Hausdorff distance. The spaces $\mathbf{M}_h=(\mathcal{M}(X),h)$ and $ \mathbf{M}_d=(\mathcal{M}(X),d)$ are asymmetric semimetric spaces. Note that the semimetric $h$ is the symmetrization of the metric $d$. On the space $\mathcal{M}=\mathcal{M}(X)$, we also consider the semilinear subspace of all convex bounded subsets $\mathcal{L}=\mathcal{L}(X)$. When equipped with the semimetric $d$ (respectively, $h$), this space defines the asymmetric semimetric semilinear space $\mathbf{L}_d=\mathbf{L}_d(X)$ (respectively, $\mathbf{L}_h=\mathbf{L}_h(X)$). The following two results can be found in [12]. Remark 11. Let $\mathcal{X}=(X,\|\,{\cdot}\,\|)$ be a seminormed linear space. Then the semimetrics of the spaces $\mathbf{L}_d$ and $ \mathbf{L}_h$ are generalized convex. In addition, if $\mathcal{X}$ is complete, then so are $ \mathbf{M}_h$ and $ \mathbf{L}_h$. We also note that $h(A,B)=h(A+C,B+C)$ and $d(A,B)=d(A+C,B+C)$, $h(\alpha A,\alpha B)=\alpha h(A,B)$, and $d(\alpha A,\alpha B)=\alpha d(A,B)$ for all $A,B,C\in \mathcal{L}$ and $\alpha\geqslant 0$. Corollary 4. Let $\mathcal{X}=(X,\|\,{\cdot}\,\|)$ be a seminormed linear space. Then and for all $A,B \in \mathcal{L}$ and $\alpha\geqslant 0$. Theorem 9. Let $X$ be a real reflexive normed space. Then each non-empty bounded subset of the space $\mathbf{L}_h=\mathbf{L}_h(X)$ admits a Chebyshev centre. Proof. Let $\mathbf{M}\subset \mathbf{L}_h$ be a non-empty bounded family of convex non-empty bounded sets $\{M_\alpha\}\subset X$. It can be assumed without loss of generality that each $M_\alpha$ is closed, and, therefore, weakly compact in $X$. Let $\{A_n\}\subset X$ be a sequence of non-empty closed bounded sets such that
$$
\begin{equation*}
h(A_n,M_\alpha)\leqslant r_n\to r:=r(\mathbf{M}):= \mathop{\smash\inf\vphantom\sup} _{A\in \mathbf{L}_h}\sup_{M_\alpha\in \mathbf{M}} h(M_\alpha,A)\quad \text{for each}\quad M_\alpha\in \mathbf{M}.
\end{equation*}
\notag
$$
Without loss of generality, passing to a subsequence if necessary, it can be assumed that $\{r_n\}$ is decreasing. Let $B_n$ be the closure of the convex hull of the set $\bigcup_{k\geqslant n}A_k$ in the space $X$ ($n\in \mathbb{N}$). Each $A_k$ ($k\geqslant n$) lies in the convex closed set $M_\alpha+B(0,r_n)$ for all $\alpha$, and hence $B_n$ ($n\in \mathbb{N}$) is contained in $M_\alpha+B(0,r_n)$ for all $\alpha$. In addition, we have
$$
\begin{equation*}
B_n+B(0,r_n)\supset A_n+B(0,r_n)\supset M_\alpha \quad \text{ for all }\alpha.
\end{equation*}
\notag
$$
The set $B:=\bigcap_{n\in \mathbb{N}}B_n$ is convex closed and non-empty (since $\{B_n\}$ is a nested sequence of weakly compact sets), and $M_\alpha+B(0,r) \supset B$ for all $\alpha$. Let us show that $B+B(0,r)\supset M_\alpha$ for all $\alpha$. Indeed, for each $M_\alpha$ and an arbitrary point $I\in M_\alpha$, there exists a point $x_n\in A_n\subset B_n$ such that $\|y-x_n\|\leqslant r_n$ ($n\in \mathbb{N}$). Each weak limit point $x$ of the sequence $\{x_n\}$ lies in each set $B_m$ ($m\in \mathbb{N}$), since $x_k\in B_m$ for all $k\geqslant m$. Hence $x\in B$, and, of course,
$$
\begin{equation*}
r\geqslant\inf\lim_{n\to\infty}\|y-x_n\|\geqslant \|y-x\|.
\end{equation*}
\notag
$$
Since $y\in M_\alpha$ is arbitrary, we have $B+B(0,r)\supset M_\alpha$ for all $\alpha$. As a result, $h(M_\alpha,B)\leqslant r$ for all $\alpha$, and, therefore, $B\in \mathbf{L}_h$ is a Chebyshev centre of the set $ \mathbf{M}$. Theorem 9 is proved. 
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\Bibitem{Tsa23} |
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