Uniform convexity in nonsymmetric spaces
An asymmetric norm on a real vector space \(X\) is a positively definite sublinear functional \(\|\cdot| :X\to \mathbb{R}_+\) (i.e., a positively homogeneous, sublinear and positively definite functional). Equivalently, \(\|\cdot|\) is the Minkowski gauge of an absorbing, radially bounded and convex subset of \(X\). An asymmetric normed space \((X,\|\cdot|)\) is a real vector space \(X\) endowed with an asymmetric norm \({\|\cdot|}\). The topology of an asymmetric normed space is \(T_1\), but not necessarily \(T_2\) (see [\textit{S.\,Cobzaş}, Functional analysis in asymmetric normed spaces. Basel: Birkhäuser (2013; Zbl 1266.46001)]). A left Cauchy sequence (also called left \(K\)-Cauchy by some authors) is a sequence \(\{x_n\}\) in \(X\) such that for every \(\varepsilon>0\) there exists \(N\in\mathbb{N}\) such that \(\|x_m-x_n|<\varepsilon\) for all \(m\ge n\ge N\). The space \(X\) is called left complete if every left Cauchy sequence is convergent. For a nonempty subset \(M\) of \(X\) and \(x\in X\), put \(\rho(x,M)=\inf\{\|y-x|: y\in M\}\), the distance from \(x\) to \(M\), and \(P_Mx= \{y\in M:\|y-x|=\rho(x,M)\}\), the metric projection. For \(x, y\in S_X\) (the unit sphere of \(X\)) and \(a\in[0,1]\), put \(\Delta(a)=\|x-ay|+ a\|y|-\|x|\). The space \(X\) is called uniformly convex if for every \(\varepsilon>0\) and \(a\in(0,1]\) there exists \(\delta>0\) such that for all \(x, y\in S_X\), \(\Delta(a) < \delta\) implies \(\|x-\mu y|\le\varepsilon\) for some \(\mu\in [1-\varepsilon,1].\) The space \(X\) is called locally uniformly convex if for all \(y\in S_X\), \(\varepsilon>0\) and \(a\in(0,1]\) there exists \(\delta>0\) such that for any \(x\in S_X\), \( \Delta(a)<\delta\) implies \(\|x-y|\le \varepsilon\). As expected, a uniformly convex space is locally uniformly convex. With these definitions, most properties of uniformly convex and locally uniformly convex normed spaces are preserved in the asymmetric case, too. For instance, if \((X,\|\cdot|)\) is a uniformly convex, left complete and \(T_2\) asymmetric normed space, then for every closed convex subset \(M\) of \(X\) and every \(x\in X\) there exists \(y\in M\) such that \(P_Mx=\{y\}\) (i.e., \(x\) has a unique nearest point in \(M\)) and for every sequence \(\{y_n\}\) in \(M\), \(\|y_n-x|\to\rho(x,M)\), \(n\to\infty\), implies \(\|y_n-y|\to 0\). As an example of uniformly convex asymmetric normed space one gives the space \(L_{p,q,\Psi}(\Omega)\), defined as follows. Let \(1<p,q<\infty\) and let \(\Psi:\mathbb{R}^2\to \mathbb{R}\) be a uniformly convex norm on \(\mathbb{R}^2\) such that \(\Psi(u,v)=\Psi(|u|,|v|)\). Then \(L_{p,q,\Psi}(\Omega)\) is the space \(L_p(\Omega)\cap L_q(\Omega)\) equipped with the asymmetric norm \(\|f|_{p,q}:=\Psi(\|f_+\|_p,\|f_-\|_q)\). Notice that another approach to smoothness and rotundity in the asymmetric setting was done via the Minkowski functionals of some rooted convex bodies in normed spaces, see [\textit{C.\,Zanco} and \textit{A.\,Zucchi}, Boll. Unione Mat. Ital., VII. Ser., B 7, No. 4, 833--855 (1993; Zbl 0804.52001)] and [\textit{V.\,Klee} et al., Stud. Math. 120, No. 3, 191--204 (1996; Zbl 0863.46004)]. \textbf{Reviewer's remarks.} In the English version, upper case and lower case letters are used in some cases to designate the same object. For instance: \(\delta(a)\) and \(\Delta(a)\), \(L_p(\Omega)\) and \(\ell_p(\Omega)\), \(\psi(u,v)\) and \(\Psi(u,v)\). In the original Russian version, these are written consistently.
scientific article; zbMATH DE number 7455556