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This article is cited in 1 scientific paper (total in 1 paper) Approximative and structural properties of sets in asymmetric 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. IntroductionAt present time, approximation in high-dimensional spaces by sets of complex structure (as a rule, nonconvex) evokes considerable interest. Of special interest here is the problem of stability of best and near-best approximants. Properties of these kind are usually important for construction of stable approximation algorithms. Theoretically, stability properties are relevant, in particular, for problems of estimation of best approximations in terms of the Alexandrov width. It frequently happens that elements of best approximations in a given set are not unique, but even in this case stable selections of operators of near-best approximations can be obtained. For elements of best approximation, it is frequently desirable to be able to single out such elements from other elements of the approximating set (such elements can be characterized in terms of solar properties of sets). Constructions of various approximation algorithms frequently call for a characterization of best approximants. In the present paper, we will be interested in how such characterization results are related to stability of best approximation. It is worth pointing out here that the definition of a sun (or a strict sun) is nothing other than an equivalent geometrical reformulation of the well-known Kolmogorov criterion for a best approximant. The above tasks can be successfully solved using the machinery of continuous selections of the operators of near-best approximation or of continuous selections of the metric projection operator. Historically, the Kolmogorov criterion was first formulated for subspaces and then carried over to the convex set setting, but later the strict suns were found to constitute the most natural class of sets for which the (generalized) Kolmogorov criterion holds. In the present paper, we will see how strict solarity implies stability of the operator of near-best approximation. In what follows, in parallel with finite-dimensional normed linear spaces, we consider finite-dimensional linear spaces equipped with asymmetric norms ‖⋅|. An asymmetric norm on a linear space X is defined by the following axioms: 1) ‖αx|=α‖x| for all α⩾0, x∈X; 2) ‖x+y|⩽‖x|+‖y| for all x,y∈X; 3) ‖x|⩾0 for all x∈X; a) ‖x|=0⇔x=0. Any asymmetric norm can also be considered as the Minkowski functional of some (in general, asymmetric) body containing the origin in its kernel. In asymmetric spaces, we will study, in particular, sets with lower semicontinuous metric projection. In parallel with asymmetric norms, one frequently considers asymmetric seminorms ‖⋅|. A seminorm is defined by the above axioms 1)–3) with axiom 3, a) replaced by the condition ‖x|=0=‖−x|⇒x=0. It is also often convenient to consider the symmetrization norm defined from an asymmetric norm ‖⋅| by ‖x‖:=max (x\in X). For a linear asymmetric normed space X=(X,{\|\cdot|}), 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. The sphere of radius r with centre x is defined by S(x,r) = \{y\in X\mid \|y-x |=r\}. The dual unit sphere in the dual space X^* of a normed linear space X is denoted by S^*. The set of all extreme functionals of S^* is denoted by \operatorname{ext}S^*. Given an arbitrary subset M of an asymmetric seminormed space X , the distance from a point y\in X to the set M is defined by \varrho(y,M) := \inf_{z\in M}\|z-y|. The set of all nearest points from M to a point x\in X is defined by P_Mx :=\{y\in M\mid \|y-x|=\varrho(x,M)\}. Given x\in X and \delta>0, we will also consider the metric \delta-projections defined by
\begin{equation*}
\begin{gathered} \, P_M^\delta x :=\{y\in M\mid \|y-x|\leqslant\varrho(x,M)+\delta\}=M\cap B(x,\varrho(x,M)+\delta), \\ \mathring{P}_M^\delta x := \{y\in M\mid \|y-x|<\varrho(x,M)+\delta\}=M\cap \mathring{B}(x,\varrho(x,M)+\delta). \end{gathered}
\end{equation*}
\notag
Definition 1. Let \varepsilon\geqslant 0 , M\subset X. We say that \varphi\colon X\to M is an additive (multiplicative) \varepsilon -selection (of the operator of near-best approximation) if, for all x\in X , (\|\varphi(x)-x|\leqslant (1+\varepsilon)\varrho(x,M), respectively). An \varepsilon -selection on a set E\subset X is defined similarly.The above formulas with \varepsilon=0 define a 0-selection (or simply a selection) of the metric projection operator. The existence of a continuous selection of near-best approximation operators is closely related to certain special structural properties of the approximating set. For example, convex subsets of normed linear spaces are known to admit a continuous additive \varepsilon-selection for all \varepsilon>0; if these sets are closed, then there also exists a continuous multiplicative \varepsilon-selection for all \varepsilon>0. This property of convex closed sets makes it possible to extend continuous mappings from such sets to those on the entire space without changing their values. An example of a nonconvex set admitting a continuous additive (multiplicative) \varepsilon-selection for all \varepsilon>0 is given by the unit sphere of any infinite-dimensional normed linear space. Note that it frequently happens that the set of best approximants is empty or disconnected — for example, Theorem 5 in [1] gives an example of a set M admitting a continuous additive \varepsilon-selection for all \varepsilon>0, but for which the set P_Mx of nearest points in M for some point x is disconnected. For various results on \varepsilon-selections for concrete function sets, see, for example, [2]–[6]. In the present paper, we will also study suns, or, more precisely, strict suns in polyhedral finite-dimensional asymmetric spaces. In particular, we will show that any strict sun in such a space admits a continuous \varepsilon-selection for any \varepsilon>0 and that the metric projection operator onto it has cell-like vales (see Corollary 1). Note that an (ordinary) sun in some polyhedral three-dimensional asymmetric space and in some polyhedral four-dimensional normed space may fail to be P-cell-like and have a continuous \varepsilon-selection for some \varepsilon>0 (see [7]). Recall that a set M is P-cell-like if, for any x\in X, the set P_M of its nearest points in M is nonempty and cell-like. Definition 2. Let \varnothing \ne M\subset X. A point x\in X\setminus M is a solar point if there exists a point y\in P_Mx (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 ray emanating from y and passing through x such that y is a nearest point in M for any point on the ray). A point x\in X\setminus M is a strict solar point if P_Mx\ne\varnothing and if each point y\in P_Mx is a luminosity point from M for x. A set M is a sun (a strict sun) if any point from X\setminus M is a solar (strict solar) point in M for x. As we have already pointed out above, any strict sun is a so-called Kolmogorov set, that is, all best approximants from this set can be characterized in terms of the Kolmogorov criterion for a best approximant (see [8]). Next, any sun is a so-called generalized Kolmogorov set, that is, its luminosity points can be characterized via the Kolmogorov criterion. For general properties of suns, see, for example, [8]–[15]. To verify the existence of continuous \varepsilon-selection of the metric projection to strict suns, we will need to show that the metric projection onto a strict sun has cell-like values, that is, a strict sun is P-cell-like. Definition 3. A compact set Y is cell-like if there exists an absolute neighbouring retract Z and an embedding i\colon Y\to Z such that the range i(Y) is contractible in any its neighbourhood U\subset Z (see [16]). Note that the intersection of a countable nested family of contractible compact sets is cell-like. It is also worth pointing out that any upper semicontinuous mapping F\colon K\to 2^Y (K\subset Y is a compact set) with cell-like values in a Banach space Y is \varepsilon-approximable (see [16]), that is, for any \varepsilon>0, there exists a continuous single-valued mapping \varphi\colon K\to Y whose graph lies in the \varepsilon-neighbourhood of the graph of F. Below, in Theorem 4 and Corollary 4, we will obtain conditions under which any left (right) Chebyshev set is a left (right) sun in asymmetric (not necessarily finite-dimensional spaces. § 2. Cell-likeness of the sets of nearest points in asymmetric polyhedral spacesGiven a point x\in X\setminus M, by \mathfrak{N}(x) we denote the set of all inclusion-maximal faces of the sphere S(x,\varrho(x,M)) such that P_Mx has nonempty intersection with the relative interior of any element of \mathfrak{N}(x). Next, given a finite-dimensional convex set V, the relative interior of V is denoted by \operatorname{int}_0V. It is easily checked that the union contains the set P_Mx.Let H^k_a be the homothety with centre a and ratio k\geqslant 0. Given nonempty subsets A, B of an asymmetric space, by h(A,B) we will denote the Hausdorff distance between A and B with respect to the symmetrization norm. A finite-dimensional asymmetric space is called polyhedral if its unit ball is the convex hull of finitely many points. Theorem 1. Let (X,\|\,{\cdot}\,|) be a finite-dimensional asymmetric polyhedral space and let M\subset X be nonempty and closed. Suppose that, for all x\in X\setminus M, the set \mathfrak{R}(x) is cell-like. Then, for any z\in X, the set P_Mz is cell-like (that is, M is P-cell-like). Proof. Let the set \mathfrak{N}(x) defined above consists of the faces \{B_\alpha(x)\}_\alpha of the sphere S(x,\varrho(x,M)).
Assume on the contrary that there exists a point x\in U such that P_Mx is not cell-like. In this case, r=\varrho(x,M)>0. Among all such points x we chose those for which the dimension of the maximal faces B_\alpha(x) is smallest possible. Then, among these points x, we will choose those for which the number of faces B_\alpha(x) of maximal dimension m_1 is smallest possible. Next, among these points, we choose the points x such that the dimension of the faces B_\alpha(x) of maximal dimension m_2<m_1 is smallest possible, and then, among these points x, we will chose those for which the number of faces B_\alpha(x) of dimension m_2 is smallest possible. The above steps are repeated as follows: we choose those x for which the dimension of the faces B_\alpha(x), \dim B_\alpha(x)<m_2, of maximal dimension m_3 is smallest possible, and then among these points x we take those for which the number of faces B_\alpha(x) of dimension m_3 is smallest possible, and so on. In this way, we will find a point x\in X for which the faces B_\alpha(x) have the dimensions m_1>m_2>\dots>m_k=:m_0 (if \dim B_\alpha(x)=m_k=m_0, then B_\alpha(x)=:B(x)), and the number of faces of dimension m_i, i=1,\dots,k-1, is smallest possible for fixed (chosen earlier) numbers m_l (l=1,\dots,i-1). Given any y\in B_\alpha=B_\alpha(x), we define
\begin{equation*}
k_\alpha(y):=\inf\{k\in [0,1]\mid \operatorname{int}_0H^k_y(B_\alpha)\cap M\neq\varnothing\}.
\end{equation*}
\notag
Let \mathcal{B}(y)=\{\alpha\mid y\in B_\alpha\}. We set
We claim that the function k\colon A\to \mathbb{R}, where A:=\mathfrak{R}(x)=\bigcup_\alpha B_\alpha, is upper semicontinuous. Consider an arbitrary point y_0\in A and a sequence \{y_n\}\subset A converging to y_0. It suffices to verify that k_0=k(y_0)\geqslant k:=\varlimsup_{n\to\infty}k(y_n). Passing to a subsequence if necessary, we may assume without loss of generality that k(y_n)\to k as n\to\infty. Since the number of faces in the family \{B_\alpha\} is finite, it can be assumed that \{y_n\}\subset B_{\alpha_0} and k_n:=k(y_n)=k_{\alpha_0}(y_n) (n\in \mathbb{N}) for some index \alpha_0. If k_{\alpha_0}(y_0) were smaller than k, then, since \operatorname{int}_0H^{k'}_{y_n}(B_{\alpha_0})\cap M=\varnothing for all n\in \mathbb{N} with k<k'<k_n, we would get that \operatorname{int}_0H^k_{y_0}(B_{\alpha_0})\cap M=\varnothing, but this contradicts the definition of k_{\alpha_0}(y_0) and the closedness of M. So, k_0=k(y_0)\geqslant k_{\alpha_0}(y_0)\geqslant k. So, the function k(\,{\cdot}\,) is upper semicontinuous. Consider the mapping F\colon A\to 2^T, where T=P_Mx\subset M, which associates with each point y\in A the set
\begin{equation*}
M\cap H^{k(y)}_{y}[B(x,r)]=P_Mx\cap H^{k(y)}_{y}[B(x,r)]=P_M(x+k(y)(y-x))
\end{equation*}
\notag
(here, r=\varrho(x,M)). Since k(\,{\cdot}\,) is upper semicontinuous, so is the mapping F(y)=P_M(x+k(y)(y-x)). There exists \alpha_0\in \mathcal{B}(y) such that k=k(y)=k_{\alpha_0}(y), and hence, \operatorname{int}_0H^k_y(B_{\alpha_0})\cap M=\varnothing. Now we impose on \alpha_0 the additional restriction that the dimension of the face B_{\alpha_0} would be the largest possible. In this case, for the point z=x+k(y)(y-x), the number of faces (of type B_{\alpha}(z)) of dimension \dim B_{\alpha_0} is now smaller (and the number of faces of larger dimension is unchanged), which means that P_Mz=F(y) is cell-like by the choice of the point x. Thus, we have shown that the upper semicontinuous mapping F maps the cell-like set A into a subset of T=P_Mx=A\cap M, F has cell-like values, and x\in F(x) for all x\in T.
We will next proceed as in the proof of Theorem 7 in [5]. The mapping f, which associates with each w from the set the value y\in A for which w=(y,u) (u\in F(y)), is a cell-like shape equivalence, since the sets T_F and A are finite-dimensional (see [17; p. 424]). As a result, T_F is cell-like. Hence, the set T_1=\{(x,y-x)\mid x\in A,\, y\in F(x)\} is also cell-like and homeomorphic to the set
\begin{equation*}
T_2=\{(y,y-x)\mid x\in A,\, y\in F(x)\}=\{(y,y-x)\mid x\in T,\, x\in F^{-1}(y)\}.
\end{equation*}
\notag
Hence T_2 is also cell-like. But its retraction T\times \{0\} via the mapping \pi(a,b)=(a,0) is not cell-like, which cannot be the case. This contraction shows that M is P-cell-like on U. Theorem 1 is proved. Let X be a finite-dimensional polyhedral space. For each proper face \mathcal{E} of the ball B(x,r), consider some point c=c(\mathcal{E}) from the relative interior \operatorname{int}_0\mathcal{E} of \mathcal{E}; this point c will be called the centre of the face \mathcal{E}. Let M\subset X be a strict sun and let a ball B(x,R) support the set M. Consider a fixed open convex set D\subset X containing the ball B(x,R). In what follows, we will consider only the balls that lie in the set D. We claim that there exists an affine plane \mathcal{L} of maximal dimension which contains P_Mx and lies in the boundary of some open cone K(x',y')\supset K(x,y), where y\in P_Mx is any nearest point for x, y'\in P_Mx'. Indeed, the affine plane \mathcal{L} is the affine hull of some face \mathcal{E} of the ball B(x,R), and now as the required support cone K(x',y') one may take the infinite union \bigcup_{k>0}H^k_c(B(x,R)) of homothetic copies of the ball B(x,R) about the centre c=c(\mathcal{E}) of the face \mathcal{E}; moreover, K(x',y')\cap M\cap D=\mathcal{L}\cap M\cap D. The face \mathcal{E}:=B(x,R)\cap \mathcal{L} and all its subfaces will be called realizable (at a point x). The class of all realizable faces (for all points x\in X\setminus M and the corresponding numbers R=\varrho(x,M)) will be denoted by \mathfrak{R}. With each realizable face \mathcal{E}, we will associate the open set W=W(\mathcal{E}) consisting of the points z\in X\setminus M for which there is a ball B(v,r)\subset D whose interior has a common point with the relative interior of \mathcal{E}, but does not intersect M, and which contains the point z in its boundary; moreover, z is the centre of the face \mathcal{E}_z:=(\mathcal{E}-x)(r/R)+v, that is, z=(c-x)(r/R)+v. Let \mathfrak{M}(\mathcal{E}) be the class of all such balls B(v,r). Note that any face \mathcal{E}\in \mathfrak{R} is obtained from the faces of the ball B(0,1) by translation and dilation of this unit ball, that is, \mathcal{E} corresponds to the face \mathcal{E}_0:=(\mathcal{E}-x)(1/R) for the ball B(0,1). Such faces \mathcal{E}_0 will be called normalized; the class of all such faces will be denoted by \mathfrak{R}_0. Since the space is polyhedral, \mathfrak{R}_0 is a finite subfamily of the family of all faces of the ball B(0,1). Let V be an arbitrary open set containing the intersection of the face \mathcal{E}\in \mathfrak{R} with the set M. Let \mathfrak{M}_0(\mathcal{E},V) be the class of balls B(v,r)\in \mathfrak{M}(\mathcal{E}) whose homothetics with respect to the corresponding points z (see above) to a ball \widehat{B} support M and \widehat{B} \cap M\subset V. Next, let n(\mathcal{E},V) be the number of normalized faces corresponding to the faces \widehat{\mathcal{E}} from \mathfrak{R}, \widehat{\mathcal{E}}\cap M\subset V, of dilations of the balls B(v,r)\in \mathfrak{M}_0(\mathcal{E},V) with respect to the corresponding point z (see above) to a ball that supports M and lies in D. There exists an open set V_0=V_0(\mathcal{E})\subset D such that the number n(\mathcal{E}):=n(\mathcal{E},V_0) is smallest possible. We may assume without loss of generality that there is \delta>0 such that, if the corresponding point z of the ball B(v,r)\in \mathfrak{M}(\mathcal{E}) lies in O_\delta(\mathcal{E}), then B(v,r)\in \mathfrak{M}_0(\mathcal{E},V_0), and, moreover, O_\delta(\mathcal{E}\cap M)\subset V_0. Let Z_0(\mathcal{E}) be the class of all points z\in O_\delta(\mathcal{E}) for the corresponding balls B(v,r)\in \mathfrak{M}_0(\mathcal{E},O_\delta(\mathcal{E} \cap M)). Lemma 1. Let M\subset X be a nonempty closed subset of an asymmetric finite-dimensional polyhedral space (X,\|\,{\cdot}\,|). Suppose that, for any number \gamma\geqslant 1, for any ball B(v,r)\subset D which supports the set M, and any point y\in P_Mv, the balls H^{\gamma}_y(B(v,r)) also support the set M. Then the intersection of any face \mathcal{E}\in \mathfrak{R} with M is cell-like. Proof. Assume the contrary. Among the faces \mathcal{E}\in \mathfrak{R} we chose a face \mathcal{E} whose intersection with M is not cell-like and the number n_0(\mathcal{E}) is smallest possible. Let n_0(\mathcal{E})=n_0(\mathcal{E},V_0). Consider an arbitrary point z\in Z_0(\mathcal{E}) \cap W(\mathcal{E}) and the corresponding ball B(v,r)\in \mathfrak{M}_0(\mathcal{E},O_\delta(\mathcal{E}\cap M)). Let B(x_0,R_0) be the ball supporting M and which is the homothetic of the ball B(v,r) with centre z and ratio k. If the face Q_z:=k(\mathcal{E}_z-z)+z were in \mathfrak{R}, then there would exist a cone K(x',y') supporting M, containing the support cone K(x_0,y_0), and whose boundary contains a translation of the affine plane \mathcal{L} (which is the homothetic of \mathcal{E} with centre c(\mathcal{E})and infinite ratio), where y_0\in P_Mx_0. Hence the interior of the cone K(x',y') would contain the face \mathcal{E}, since the interior of the ball B(v,r)\subset B(x_0,r_0) intersects the relative interior of \mathcal{E}, but this,. however, is impossible, because \mathcal{E}\cap M\ne \varnothing.
So, any face Q_z is not realizable. For sufficiently small \delta>0, it can be assumed that, for any realizable face P of a ball \widehat{B}_z which supports M and which is a homothetic of the ball B(v,r)\in \mathfrak{M}_0(\mathcal{E},O_\delta(\mathcal{E})) about the corresponding point z, the intersection P\cap M lies in V_0. Since the faces from \mathfrak{R} corresponding to the normalized face \mathcal{E} are already excluded, we have n_0(P_z)<n_0(\mathcal{E}). Therefore, by the assumption, for any z\in O_\delta(\mathcal{E}), the intersection of the support ball \widehat{B}_z with M is a cell-like set. Let A:=\mathcal{E}\cap M=\mathcal{E}\cap P_Mx. For sufficiently small \varepsilon\in (0,1), there exist k=k(\varepsilon)>1 and \delta=\delta(\varepsilon)>0 such that H^k_c(\mathcal{E})\cap A\subset O_\varepsilon(A) (recall that c=c(\mathcal{E}) is the centre of the face \mathcal{E}), and, moreover, O_{2\delta}(A)\cap \mathcal{L}\subset \operatorname{int}_0(H^k_s(\mathcal{E})). There exists a point t\in (x,c) lying so close to x that \widehat{B}\cap M\subset O_\varepsilon(A) and O_\delta(A)\cap \mathcal{L}\subset \operatorname{int} B(t,kR), where \widehat{B}:=B(t,kR). For each point s \in \mathcal{E}, we set k(s):=\inf\{k \in [0,1]\mid \operatorname{int}_0(H^k_c(\widehat{B})) \cap M\neq \varnothing\}. According to [6], the mapping F(s):=H^{k(s)}_s(\widehat{B})\cap M is upper semicontinuous. Further, by the above, we may assume that the point t\in (x,c) is so close to x and the number k=k(\varepsilon) is so close to 1 that the set F(s) lies in O_\varepsilon(A) and is cell-like. Hence, there exists a continuous mapping \varphi\colon \mathcal{E}\to O_\varepsilon(A) such that (s,\varphi(s))\in O_\Delta(\Gamma) for all s\in \mathcal{E} and arbitrarily small \Delta\in (0,\varepsilon/3), where \Gamma = \{(p,F(p))\mid p \in \mathcal{E}\} is the graph of F, O_\Delta(\Gamma) = \{(a,b)\mid \inf_{c\in \mathcal{E}}\widehat{\varrho}((a,b),(c,F(c)))\,{<}\,\Delta\}, and \widehat{\varrho}((a,b),(c,F(c)))=\|a-c|+ \varrho(b,F(c)). We claim that \varphi(s)\in O_{2\varepsilon}(A) for all s \in \mathcal{E} and \|\varphi(s)- s|<3\varepsilon for all s \in A. Indeed, there exists a sufficiently small \Delta such that h(F(s'),F(s))=h(\{s'\},F(s))<\varepsilon, whenever s\in \mathcal{E}, s'\in A, \|s-s'|<2\Delta. Therefore, h(\{s\},F(s))< 2\varepsilon. Hence, h(\{\widetilde{s}\},F(\widetilde{s}))\leqslant 2\varepsilon whenever s\in O_\Delta(A), \widetilde{s}\in \mathcal{E}, \|s-\widetilde{s}|<\Delta. We have for all s\in \mathcal{E} and some appropriate s', and hence \|s'-s|<\Delta, \varrho(\varphi(s),F(s'))<\Delta, and \|\varphi(s)-s|\leqslant \|s'-s|+h(\{s'\},F(s'))+\varrho(\varphi(s),F(s'))<3\Delta+2\varepsilon<3\varepsilon for all s\in A.Consider the continuous function
\begin{equation*}
\Psi(s)=\begin{cases} \varphi(s), &s\in \mathcal{E}\setminus O_{2\varepsilon}(A), \\ s(1-\tau(s))+\tau(s)\varphi(s), &s\in O_{2\varepsilon}(A), \end{cases}
\end{equation*}
\notag
where \tau(s):=\min\{1,\varrho(x,A)/(2\varepsilon)\}. We have \Psi\in C(\mathcal{E}), \|\Psi(s)-s|\leqslant \|\varphi(s)- s| \leqslant \operatorname{diam}\mathcal{E}+2\varepsilon on \mathcal{E} and \Psi(s)=s on A. Let the mapping \chi\colon \mathcal{E}\to O_{3\varepsilon}(A)\times O_{3\varepsilon}(0) be defined by \chi(s)=(\Psi(s),\varepsilon/(4(\operatorname{diam}\mathcal{E}+1))(\Psi(s)-s)). This mapping is a homeomorphism between the sets \mathcal{E} and \chi( \mathcal{E}). Therefore, \chi(\mathcal{E}) is a contractible set which contains the set A\times \{0\}\subset O_{3\varepsilon}(A)\times O_{3\varepsilon}(0). Since \varepsilon>0 is arbitrary, the set A is cell-like, which, however, contradicts the construction of the face \mathcal{E}. This proves Lemma 1. The following result is a direct consequence of this lemma. Corollary 1. Any strict sun M in any asymmetric finite-dimensional polyhedral space (X,\|\,{\cdot}\,|) is P-cell-like, that is, P_Mx is cell-like for each x\in X. Note that Corollary 2 in [6] implies that, under the hypotheses of Corollary 1, the set M admits a continuous \varepsilon-selection for all \varepsilon>0. § 3. Suns in asymmetric spacesDefinition 4. The subbase of balls \mathring{B}(x,r)=\{y\in X\mid \|y-x|< r\} (respectively, \mathring{B}^-(x,r)=\{y\in X\mid \|x- y|< r\}) in an asymmetric linear space X=(X,\|\,{\cdot}\,|) generates the topology, which we will call the right (left) topology. The elements of the right (left) topology are the right- (left-) open sets. To indicate the topology under consideration, we will use the subscripts ‘+” or ‘-’ as follows: given a set M\subset X we write M_+ (respectively, M_-) if M is equipped with the right (left) topology; the notation M^s means that M is equipped with the symmetrization topology (the topology generated by the symmetrization norm). A subset K of an asymmetric linear space X=(X,\|\,{\cdot}\,|) is right- (left-) compact if any cover of M by right- (left-) open sets contains a finite subcover. A set will be said to be strongly compact if it is compact in the symmetrization topology; this is equivalent to saying that the set is compact both in the right and left topologies. Definition 5. We say that a set E=\{x_i\}_{i=1}^{N} is a finite right (left) \varepsilon-net for a subset K of an asymmetric linear space X = (X,\|\,{\cdot}\,|) if K\,{\subset} \bigcup_{i=1}^{N}\mathring{B}(x_i,\varepsilon) (respectively, K\subset \bigcup_{i=1}^{N}\mathring{B}^-(x_i,\varepsilon)). Corollary 2. If a set K in an asymmetric linear space X=(X,\|\,{\cdot}\,|) is right- (left-) compact, then K admits a finite right (left) \varepsilon-net consisting of points of K. Proof. The covers \{\mathring{B}(x,\varepsilon)\}_{x\in K} (\{\mathring{B}^-(x,\varepsilon)\}_{x\in K}) admit finite subcovers; the centres of these balls form the required finite \varepsilon-net for K. The convex hull of a set M will be denoted by \operatorname{conv} M. Remark 1. If E=\{x_i\}_{i=1}^{N}\subset K is a finite right (left) \varepsilon-net for K in an asymmetric linear space X=(X,\|\,{\cdot}\,|), then \operatorname{conv} E is a finite-dimensional compact set, and, in addition, \operatorname{conv} K\subset \operatorname{conv} E+\mathring{B}(0,\varepsilon) (\operatorname{conv} K\subset \operatorname{conv} E+\mathring{B}^-(0,\varepsilon)). Definition 6. Let \mathcal{L}_n be the class of all planes of dimension \leqslant n\in \mathbb{Z}_+. The right (left) Kolmogorov width of order n is defined by
\begin{equation*}
\begin{gathered} \, d_n(K,X)=\inf\{\varepsilon\geqslant 0\mid K\subset L+\mathring{B}(0,\varepsilon)\ \text{for some} \ \ L\in \mathcal{L}_n\} \\ \bigl(d_n^-(K,X)=\inf\{\varepsilon\geqslant 0\mid K\subset L+\mathring{B}^-(0,\varepsilon)\ \text{for some} \ L\in \mathcal{L}_n\}\bigr). \end{gathered}
\end{equation*}
\notag
The following result is a direct consequence of Corollary 2 and Remark 1. Corollary 3. Let K be a right- (left-) compact set in an asymmetric linear space X=(X,\|\,{\cdot}\,|). Then d_n(K,X) \to 0 (d_n^-(K,X)\to 0) as n\to\infty. Remark 2. If K is a right- (left-) compact subset of an asymmetric linear space X=(X,\|\,{\cdot}\,|), then, for any sequence \{x_k\}\subset K, there exist a point x\in K and a subsequence \{x_{k_n}\} such that \|x_{k_n}-x|\to 0 (\|x-x_{k_n}|\to 0) as n\to\infty. The proofs of both these results are similar, and so we will verify only the first claim. Assume on the contrary that, for each z\in K, there is \varepsilon(z)>0 such that the ball \mathring{B}(z,\varepsilon(z)) contains only a finite number of elements of the sequence \{x_k\}. Then, choosing a finite subcover from the cover \{\mathring{B}(z,\varepsilon(z))\}_{z\in K}, we find that K contains only a finite number of elements from \{x_k\}, which is impossible. Hence there exists a point x\in K such that any neighbourhood of x contains an infinite number of elements of \{x_k\}. Now by induction, for each n\in \mathbb{N}, n\geqslant 2, there exists k_n > k_{n-1} such that x_{k_n}\in \mathring{B}(x,1/n). This shows that \|x_{k_n}-x|<1/n\to 0 as n\to\infty. The right-distance from a point x to a subset M of an asymmetric linear space X=(X,\|\,{\cdot}\,|) is defined by \varrho^+(x,M)=\varrho(x,M):=\inf_{z\in M}\|z-y|, the left distance is defined by Definition 7. An asymmetric space X=(X,\|\,{\cdot}\,|) is called right- (left-) complete if, for any sequence \{x_n\}\subset X, the condition that, for any \varepsilon>0, exists N\in \mathbb{N} such that \|x_m-x_n|<\varepsilon for all m\geqslant n\geqslant N, implies that there exists a point x\in X such that \|x-x_n|\to 0 (\|x_n-x|\to 0) as n\to\infty. For brevity, a right-complete space will be called a complete space. Similarly, one says that a subset of an asymmetric spaces is right- (left-) complete if each Cauchy sequence from this set right- (left-) converges to an element of this set. Cobzaş (see [18; Theorem 1.6]) proved that if a set m in a left-complete asymmetric linear space X contains, for any \varepsilon>0, a finite right \varepsilon-net, then the closure of the convex hull of M is right-compact. Theorem 2. Let K be a right-compact convex subset of an asymmetric linear space X=(X,\|\,{\cdot}\,|). Then \inf_{x\in K}\|f(x)-x|=0 for any f\in C(K^s,K_+). If, in addition, f\in C( K_+,K_-), then f has a fixed point x\in K, that is, f(x)=x. Proof. Given an arbitrary \varepsilon>0, let T=T_\varepsilon be the convex hull of some finite \varepsilon-net for K. Note that the restriction of the right or left topology to T is the topology \tau generated by the symmetrization norm. We set
\begin{equation*}
\Phi(x):=\{t\in T\mid t\in \mathring{B}^-(x,\varepsilon)\}.
\end{equation*}
\notag
We claim that \Phi is lower semicontinuous in the sense that the pre-image is equipped with the right topology, and the image, with the topology \tau. Indeed, if \|z_n-z|\to 0 as n\to\infty and y\in \Phi(z), then there exists \delta\in (0,\varepsilon) such that \|z-y|<\delta, and \|z_n-y|\leqslant \|z_n-z|+\|z-y|< \|z_n-z|+\delta<\varepsilon starting from some n_0. Therefore, y\in \Phi(z_n) for all n>n_0.
Hence the mapping \Psi:=\Phi\circ f is \tau-lower semicontinuous on T (both the image and pre-image are equipped with the topology \tau); the values of \Psi are nonempty open convex subsets of T. By the Michael selection theorem [19], the mapping \Psi admits a continuous selection \varphi\colon T\to T. Therefore, there exists a point x_\varepsilon=\varphi(x_\varepsilon)\in \Psi(x_\varepsilon). Since K\subset T+\mathring{B}^-(0,\varepsilon), we have \varrho^-(f(x_\varepsilon),T)< \varepsilon (this follows from the definition of \Phi) and \|f(x_\varepsilon)-x_\varepsilon|<\varepsilon. As a result, \inf_{x\in K}\|f(x)-x|=0. Assume in addition that f\in C(K_+,K_-). Let \{\varepsilon_n\} be a positive null sequence and let T_n=T_{\varepsilon_n} be the convex hull of some finite \varepsilon_n-net for K, n\in \mathbb{N}. We can find a point x_n\in K such that \|f(x_n)-x_n|<\varepsilon_n, n\in \mathbb{N}. There exist a subsequence \{x_{k_n}\} and a point x\in K such that \|x_{k_n}-x|\to 0 as n\to\infty. Hence
\begin{equation*}
\|f(x)-x|\leqslant \|x_{k_n}-x|+\|f(x_{k_n})-x_{k_n}|+\|f(x)-f(x_{k_n})|\to 0\quad (n\to\infty),
\end{equation*}
\notag
which gives f(x)=x, thereby proving the lemma. Remark 3. In any asymmetric space X in which the ball B(0,1) is a closed set, any right- (left-) compact set K\subset X is a paracompact set. From this fact, a standard trick produces, for any nonempty convex subset M of X, a continuous \varepsilon-selection \varphi\in C(K_+,M_+) (\varphi\in C(K_-,M_-)) for all \varepsilon>0, that is, \|\varphi(x)-x|<\varrho^+(x,M)+\varepsilon (respectively, \|x-\varphi(x)|<\varrho^-(x,M)+\varepsilon) for each x\in K. Remark 4. A similar argument establishes the following result. Let K be a non-empty left-compact convex subset of an asymmetric linear space X=(X,\|\,{\cdot}\,|). Then \inf_{x\in K}\|x- f(x)|= 0 for any mapping f\in C(K^s,K_-). If, in addition, f\in C(K_-,K_+), then f has a fixed point x\in K, that is, f(x)=x. Remark 5. Theorem 2 actually shows that if a (not necessarily compact) set K has a finite right (left) \varepsilon-net, then, for any mapping f\in C(K^s,K_+) (f\in C(K^s,K_-)), there exists a point x\in K such that \|f(x)-x|<\varepsilon (\|x-f(x)|<\varepsilon). Next, to verify the existence of a fixed point in this lemma, it suffices, in place of the condition that K is a right compact set, to suppose that any \{x_n\}\subset K has a subsequence \{x_{n_k}\} such that \|x_{n_k}-x|\to 0 as k\to\infty for some point x\in K. Remark 6. Note that if a space X=(X,\|\,{\cdot}\,|) is right- (left-) complete, then the mirror asymmetric space X^-=(X,\|-\cdot|) is also inversely right- (left-) complete. Remark 7. A subset N of an asymmetric space X=(X,\|\,{\cdot}\,|) is closed (in the topology generated by open balls) if y\in N whenever \{y_n\}\subset N, \|y_n-y|\to 0 as n\to\infty. Remark 8. Definition 7 can be given not only for linear spaces X with asymmetric norm, but also for spaces X with asymmetric seminorm. Definition 8. A set M in an asymmetric space X=(X,\|\,{\cdot}\,|) will be said to be right- (left-) boundedly compact if its intersection with any right ball B(x,r) (left ball B^-(x,r)) is right- (left-) compact. The left and right metric projection operators are defined by
\begin{equation*}
P_M^-(x):=\{y\in M\mid \|x-y|=\varrho^-(x,M)\}, \ \ \ P_M^+(x):=\{y\in M\mid \|y-x|=\varrho^+(x,M)\},
\end{equation*}
\notag
where x\in X, M\subset X. We also set for brevity P_M(x)=P_M^+(x). Proposition 1. Let M be a left- (right-) boundedly compact subset of an asymmetric linear space X = (X,\|\,{\cdot}\,|), and let the distance function \varrho^-(\,{\cdot}\,,M) (\varrho^+(\,{\cdot}\,,M)) be left-left (right-right) continuous at a point x\in X (that is, both the pre-image and the image are equipped with the left (right) topology). Then the metric projection operator P^-_M (P_M) is left-left (right-right) upper semicontinuous. Proof. The proofs of both these results are similar, and so we will verify only the first claim. Let us show that, for any sequence \{x_n\}\subset X, \|x_n-x|\to 0 as n\to\infty, and any \varepsilon>0, there exists N\in \mathbb{N} such that
\begin{equation*}
P^-_M(x_n)\subset O_\varepsilon^-(P^-_M(x ))\quad\text{for all}\quad n\geqslant N.
\end{equation*}
\notag
Assume on the contrary that there exists a point y_n\in K_n:=P^-_M(x_n)\setminus O_\varepsilon^-(P^-_M(x )) for an infinite number of n’s. Passing to a subsequence if necessary, we may assume without loss of generality that y_n\in K_n for all n\in \mathbb{N}. Since \varrho^-(x_n,M)\to \varrho^-(x,M), we have
\begin{equation*}
\|x-y_n|\leqslant \|x_n-x\|+\|x_n-y_n|\leqslant R_n+ \varrho^-(x,M)\to \varrho^-(x,M)\quad (n\to\infty),
\end{equation*}
\notag
where \{R_n\} is some positive null sequence. The set \widetilde{M}_n:=(M\cap B^-(x,R_n))\setminus O_\varepsilon^-(P^-_M(x )) is left-compact, and hence the sequence \{y_m\}_{m\geqslant n}\subset \widetilde{M}_n has a subsequence converging in the left topology to an element from \widetilde{M}_n. Next, \{\widetilde{M}_n\} is a nested sequence of nonempty compact sets, and hence the set M_0:=\bigcap_{n}\widetilde{M}_n is nonempty. Hence there exists a point y_0\in M_0\subset P^-_M(x)\setminus O_\varepsilon^-(P^-_M(x )), which cannot be the case. This proves Proposition 1. Theorem 3. Let M be a left- (right-) boundedly compact subset of an asymmetric linear space X=(X,\|\,{\cdot}\,|). Then the metric projection P^-_M (P_M) is upper semicontinuous in the sense that the pre-image is equipped with the topology of the symmetrization norm, and the image is equipped with the left (right) topology. Proof. The proofs of both these results are similar, and so we will verify only the first claim. Let us first show that the function \varrho^-(\,{\cdot}\,,M) is continuous with respect to the symmetrization norm. Indeed, for all x,y\in X, z\in M, Hence \varrho^-(x,M) \leqslant \|x-y|+\varrho^-(y,M), and so, \varrho^-(x,M)-\varrho^-(y,M) \leqslant \|x-y|. Similarly, \varrho^-(y,M)- \varrho^-(x,M)\leqslant \|y-x|, and, therefore, |\varrho^-(x,M)-\varrho^-(y,M)| \leqslant \|x-y\|. It follows that the function \varrho^-(\,{\cdot}\,,M) is continuous with respect to the symmetrization norm. Note also that if \|x_n-x\|\to 0 as n\to\infty, then
Now let us show that, for any sequence \{x_n\}\subset X, \|x_n-x\|\to 0 as n\to\infty, and any number \varepsilon>0, there exists N\in \mathbb{N} such that
\begin{equation*}
P^-_M(x_n)\subset O_\varepsilon^-(P^-_M(x ))\quad\text{for all}\quad n\geqslant N.
\end{equation*}
\notag
Assume on the contrary that there exists a point y_n\in K_n:=P^-_M(x_n)\setminus O_\varepsilon^-(P^-_M(x )) for an infinite number of n’s. Passing to a subsequence if necessary we may assume without loss of generality that y_n\in K_n for all n\in \mathbb{N}. Since
\begin{equation*}
|\varrho^-(x_n,M)-\varrho^-(x,M)| \leqslant \|x_n-x\|\quad\text{and}\quad \|x_n-y_n|=\varrho^-(x_n,M)\to \varrho^-(x,M),
\end{equation*}
\notag
we have
\begin{equation*}
\|x-y_n|\leqslant \|x_n-x\|+\|x_n-y_n|\leqslant R_n+ \varrho^-(x,M)\to \varrho^-(x,M)\quad (n\to\infty),
\end{equation*}
\notag
where \{R_n\} is some positive null sequence. Since the set \widetilde{M}_n:=(M\cap B^-(x,R_n))\setminus O_\varepsilon^-(P^-_M(x )) is left-compact, the sequence \{y_m\}_{m\geqslant n}\subset \widetilde{M}_n has a subsequence left-converging to some point from \widetilde{M}_n. Next, \{\widetilde{M}_n\} is a nested sequence of nonempty compact sets, and hence M_0:=\bigcap_{n}\widetilde{M}_n is nonempty. Hence there exists a point y_0\in M_0\subset P^-_M(x)\setminus O_\varepsilon^-(P^-_M(x )), which cannot be the case. Now the required result follows. Theorem 3 is proved. Definition 9. Let M be a nonempty subset of an asymmetric linear space X=(X,\|\,{\cdot}\,|). A point x\in X\setminus M is a right (left) solar point from M for x if there exists a point y\in P^+_Mx\ne \varnothing (y\in P^-_Mx\ne \varnothing) (called a right (left) luminosity point) such that y\in P^+_M((1-\lambda)y+\lambda x) (y\in P^-_M((1-\lambda)y+\lambda x)) for all \lambda\geqslant 0. A point x\in X\setminus M is called a right (left) strict solar point if P^+_Mx\ne\varnothing (P^-_Mx\ne\varnothing) and each point y\in P_Mx is a right (left) luminosity point. If any point X\setminus M is a right (left) solar (strict solar) point from M for x, then M is called a right (left) sun (strict sun). Definition 10. A subset M of an asymmetric space X=(X,\|\,{\cdot}\,|) is called a right (left) \gamma-sun if, for any \delta>0, R>\delta, and x\in X\setminus M, there exists a point z\in X, \|x-z|=R (\|z-x|=R), such that Definition 11. A a nonempty subset M of an asymmetric linear space X=(X,\|\,{\cdot}\,|) is called a right (left) Chebyshev set if P^+_Mx (P^-_Mx) is a singleton for any x\in X. Theorem 4. Let M be a left- (right-) boundedly compact left (right) Chebyshev set in an asymmetric linear space X=(X,\|\,{\cdot}\,|) in which the ball B(0,1) is closed. Then M is a left (right) \gamma-sun. If, in addition, the intersection of M with any left (right) ball is right- (left-) compact and the mapping P^-_M (respectively, P^+_M) is left-left (right-right) continuous (that is, both the pre-image and the image are equipped with the left (right) topology), then M is a left (right) sun. Proof. The proofs of both results are similar, and so we will verify only the first claim. Consider an arbitrary point x_0\in X\setminus M. We may assume without loss of generality that x_0=0. In [20] it was pointed out that Proposition 1.1.8 of [18] implies that the closedness of the ball B(0,1) is equivalent to the continuity of the asymmetric norm. Theorem 3 and the hypotheses of the theorem imply that P^-_M is a single-valued continuous mapping in the sense that the pre-image is equipped with the topology of the symmetrization norm and the image is equipped with the left topology. The mapping \varphi(x):=-x is left-right continuous (in the pre-image one considers the left topology, and in the image, the right topology). The mapping \psi(x):=Rx/\|x|, where R>\varrho^-(0,M) is an arbitrary number, is right-right continuous (the right topology is considered both in the pre-image and in the image). There exists R_0>0 such that P^-_M(B^-(0,R))\subset B^-(0,R_0). The set M\cap B^-(0,R_0) is compact in the left topology, and hence E:=\psi\,\circ\, \varphi(M\cap B^-(0,R_0))\subset B^-(0,R) is a compact set in the right topology. On the set \operatorname{conv}E the mapping \Phi:=\psi\,\circ \,\varphi\,\circ\, P^-_M is a continuous mapping in the sense that the pre-image is equipped with the topology of the symmetrization norm and the image is equipped with the right topology. By Remarks 1 and 5, for any number \varepsilon>0, there exists a point x_\varepsilon\in \operatorname{conv}E z_\varepsilon:=\Phi(x_\varepsilon) and \|z_\varepsilon-x_\varepsilon|\leqslant \varepsilon. Let y=P^-_M(x_\varepsilon). Hence
\begin{equation*}
\begin{gathered} \, \|z_\varepsilon-y|=\|0-y|+\|z_\varepsilon-0| =\|0-y|+R\geqslant \varrho^-(0,M)+R, \\ \varrho^-(0,M)+R\leqslant \|z_\varepsilon-y|\leqslant \|x_\varepsilon-y|+ \|z_\varepsilon-x_\varepsilon|=\varrho^-(x_\varepsilon,M)+\|z_\varepsilon-x_\varepsilon|, \end{gathered}
\end{equation*}
\notag
and, therefore,
\begin{equation*}
R-\varepsilon\leqslant R-\|z_\varepsilon-x_\varepsilon|\leqslant \varrho^-(x_\varepsilon,M)-\varrho^-(0,M).
\end{equation*}
\notag
Hence M is a left \gamma-sun.
Assume in addition that P^-_M is a left-left continuous mapping (both the pre-image and the image are equipped with the left topology) and that the intersection of M with any left ball is right-compact. In this case, \psi\circ \varphi is a right-left continuous mapping (the pre-image is equipped with the right topology, and the image, with the left topology). Hence E:=\psi\circ \varphi(M\cap B^-(0,R)) is a compact set in the left topology. For a positive null sequence \{\varepsilon_k\}, consider the corresponding sequences \{x_k:=x_{\varepsilon_k}\} and \{z_k:=z_{\varepsilon_k}=\Phi(x_k)\}. In this case, R-\varepsilon_k\leqslant \varrho^-(x_k,M)-\varrho^-(0,M) (k\in \mathbb{N}) and \|z_k-x_k|\leqslant \varepsilon_k\to 0 (k\to\infty). Since E is a compact set in the left topology, some subsequence of the sequence \{z_k\} converges to some point z\in E with respect to the left topology. Passing if necessary to a subsequence, we may assume without loss of generality that \|z-z_k|\to 0 as k\to\infty. As a result, \|z-x_k|\leqslant\|z_k-x_k|+\|z-z_k|\to 0 as k\to\infty and
\begin{equation*}
\begin{aligned} \, R-\varepsilon_k &\leqslant \varrho^-(x_k,M)-\varrho^-(0,M)\leqslant \varrho^-(z,M)+\|z-x_k|-\varrho^-(0,M) \\ &\leqslant \varrho^-(z,M)+\|z_k-x_k|+\|z-z_k|-\varrho^-(0,M)\quad (k\in \mathbb{N}). \end{aligned}
\end{equation*}
\notag
Making k\to\infty, we find that Next, \|z-0|=\lim_{k\to\infty}\|z_k|=R, since the norm \|\,{\cdot}\,| is continuous. Now for the point y_0=P^-_M(0) we have
\begin{equation*}
\varrho^-(0,M)+R\leqslant\varrho^-(z,M)\leqslant\|z-y_0|\leqslant \|0-y_0|+\|z-0|\leqslant \varrho^-(0,M)+R.
\end{equation*}
\notag
It follows that B^-(0,\varrho^-(0,M))\subset B^-(0,\varrho^-(0,M)+R)=B^-(0,\varrho^-(z,M)), and since M is a left Chebyshev set, we find that y_0=P^-_M(0) is a unique nearest point also for the point z. In addition, we have P^-_M(x_k) \to P^-_M(z) as k\to\infty, and further, since 0\in [z_k,P^-_M(x_k)], we find that 0\in [z,y_0]. As a result, it can be easily shown that y_0 is a nearest point in M for any point from the interval [z,y_0]. Finally, since R is arbitrary, y_0 is a nearest point in M for any points from the ray \{y_0+t(0-y_0)\mid t\geqslant 0\}. Thus, x_0=0 is a left-solar point for M, and therefore, M is a sun, since x_0 is arbitrary. This proves the theorem. Remark 9. The second conclusion of Theorem 4 remains valid if its hypothesis that the metric projection P^-_M (respectively, P^+_M) is left-left (respectively, right-right) continuous is replaced by strict convexity of X (a space is strictly convex if its unit sphere does not contain nondegenerate intervals). Corollary 4. Let M be a left- (right-) boundedly strongly compact left (right) Chebyshev set in an asymmetric strictly convex linear space X=(X,\|\,{\cdot}\,|) in which the ball B(0,1) is closed. Then M is a left (right) sun. 
Citation in format AMSBIB
\Bibitem{Tsa22}
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