Convexity of \(\delta\)-suns and \(\gamma\)-suns in asymmetric spaces (Q6585814)

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)]). Balls are defined by \(B(x,r)=\{y\in X: \|y-x|\le r\}\) -- a ``closed'' ball, and \(\overset{\circ}{B}(x,r)=\{y\in X: \|y-x|< r\}\) -- 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 also topologically open but closed balls need not be topologically closed. The functional \(\|\cdot\|_{\mathrm{sym}}\) defined by \(\|x\|_{\mathrm{sym}}=\max\{\|x|,\|{-}x|\}\) is a norm on \(X\). The asymmetric normed space \((X,\|\cdot|)\) is called symmetrizable if the topologies generated by \(\|\cdot|\) and \(\|\cdot\|_{\mathrm{sym}}\) agree.\N\NSpheres are defined by \(S(x,r)=\{y\in X: \|y-x|=r\}\). The unit sphere \(S(0,1)\) is denoted by \(S\) and that in the dual space \(X^*\) by \(S^*\).\N\NThe distance from a point \(x\in X\) to a nonempty subset \(M\) is defined by \(\rho(x,M)=\inf\{\|y-x|:y\in M\}\). The 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 \(B(x,r)\) 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 \(B^-(x,r)=\{y\in X:\|x-y|\le r\}\).\N\NIf the asymmetric normed space \((X.\|\cdot|)\) is right complete and symmetrizable, then each closed right \(\delta\)-sun is a right \(\gamma\)-sun (Theorem 2.A in the paper -- proved in [\textit{I. G. Tsar'kov}, Izv. Math. 88, No. 2, 369--388 (2024; Zbl 1537.41025); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 88, No. 2, 184--205 (2024)]).\N\NThe paper is concerned with the convexity properties of these suns. The author presents two such instances. Every closed right \(\gamma\)-sun is convex provided that:\N\begin{itemize}\N\item the unit sphere \(S^*\) does not contain nondegenerate intervals (Corollary 2.1), or\N\item the unit sphere \(S\) of \(X\) is smooth and every functional in \(X^*\) attains its norm on the unit ball \(B(0,1)\) (Corollary 2.3).\N\end{itemize}\N\NOne says that the unit sphere \(S\) is smooth if for every \(x\in S\) there exists a unique \(x^*\in S^*\) such that \(x^*(x)=1.\)\N\NThe case of asymmetric normed cones is treated in Section 3 (the last) of the paper.