Relations between various types of suns in asymmetric spaces (Q6624458)

An asymmetric norm is a positively definite, positively homogeneous and subadditive functional \(\|\cdot|\) on a real linear space \(X\), or on a cone \(X\) (see, for instance, [\textit{Ş. Cobzaş}, Functional analysis in asymmetric normed spaces. Basel: Birkhäuser (2013; Zbl 1266.46001)]).\N\NBalls are defined by \(B(x,r)=\{y\in X: \|y-x|\le r\}\) -- a ``closed'' ball, and \({B}^\circ(x,r)=\{y\in X: \|y-x|< r\}\) -- an open ball. The topology is introduced, as usual, through the open balls forming a subbase. Notice that this topology is only \(T_1\) (not necessarily Hausdorff), the open balls are topologically open, but closed balls need not be topologically closed. The left balls are defined correspondingly, by replacing \(\|y-x|\) with \(\|x-y|\). The functional \(\|\cdot\|_{\mathrm{sym}}\) defined by \(\|x\|_{\mathrm{sym}}=\max\{\|x|,\|-x|\}\) is a norm on \(X\).\N\NSpheres are defined by \(S(x,r)=\{y\in X: \|y-x|=r\}\) and left spheres by \(S^-(x,r)=\{y\in X: \|x-y|=r\}\). The unit sphere \(S(0,1)\) is denoted by \(S\).\N\NThe distance from a point \(x\in X\) to a nonempty subset \(M\) is defined by \(\rho(x,M)=\inf\{\|z-x|:z\in M\}\) and the left distance by \(\rho^-(x,M)=\inf\{\|x-z|:z\in M\}\). The corresponding metric projections are denoted by \(P_Mx\) and \(P_M^-x,\) respectively.\N\NThe set \(M\) is called a right \(\delta\)-sun if for every \(x\in X\) with \(\rho(x,M)>0\) there exists a sequence \(\{x_n\}\) in \(X\) such that \(\|x_n-x|\to 0\) and \(\big(\rho(x_n,M)-\rho(x,M)\big)/\|x_n-x|\to 1,\) as \(n\to\infty\). Left \(\delta\)-suns are defined by replacing \(\|x_n-x|\) with \(\|x-x_n|\).\N\NThe set \(M\) is called a right \(\gamma\)-sun if for every \(x\in X\) with \(r:=\rho(x,M)>0\) and every \(\delta>0\), the ball \(B(x,r-\delta)\) is contained in a ball \(B(z,R)\) for some \(z\in X\) and an arbitrarily large radius \(R>0\). The notion of left \(\gamma\)-sun is obtained by considering left balls.\N\NA luminosity point for \(x\in X\setminus M\) is a point \(y\in P_Mx\) such that \(y\in P_m(y+\lambda(x-y))\) for all \(\lambda\ge 0,\) i.e. \(y\) is a nearest point for all points on the ray \(y+\mathbb{R}_+(x-y).\) The point \(x\) is called a solar point if there exists a luminosity point for \(x\) in \(P_Mx\), and a strict solar point if \(P_Mx\ne\emptyset\) and every point in \(P_Mx\) is a luminosity point for \(x\).\N\NIn a locally uniformly convex asymmetric normed space every proximinal \(\gamma\)-sun is a Chebyshev sun (see [\textit{I. G. Tsar'kov}, Sb. Math. 213, No. 10, 1444--1469 (2022; Zbl 1531.41033); translation from Mat. Sb. 213, No. 10, 139--166 (2022)]).\N\NWe quote from Introduction: ``In the present paper, we show that if \(P_M^-\) is Hausdorff lower semicontinuous on some neighborhood of a point of inverse left approximative compactness \(x\in X \setminus M\) for \(M\), then \(x\) is a left \(\delta\)-solar point for \(M\) (Theorem 1). Next, we will establish a link between left- (right inverse) \(f\)-suns and left (right) \(f\)-suns (Theorems 2 and 3). Further, we will obtain conditions on uniformly convex spaces and sets under which such sets are proximinal (Theorems 4.7).''