Cosuns in \(\ell ^ p(n)\), \(1\leq p<\infty \).
For a nonempty subset K of a normed space X and \(x\in X\) put \(R_ K(x)=\{k\in K:\| k-k'\| \leq \| x-k'\|\), \(\forall k'\in K\}\) and \(A_ K(x)=\{z\in X:\| z-k\| \leq \| z+\lambda (x-z)-k\|\), \(\forall k\in K\), \(\forall \lambda >0\}\). The elements of \(R_ K(x)\) (if exist) are called elements of best coapproximation of x in K. The set K is called coproximinal if \(R_ K(x)\neq \emptyset\), \(\forall x\in X\), and a cosun if \(A_ K(x)\cap K\neq \emptyset,\) \(\forall x\in X\). Every cosun is a coproximinal set and if X is finite dimensional, every coproximinal set is a cosun. The coproximinal sets are also related to the optimal sets in the sense of \textit{B. Beauzamy} and \textit{B. Maurey} [J. Funct. Anal. 24, 107-139 (1977; Zbl 0344.46049)]. The paper is devoted mainly to characterizations of cosuns in \(\ell^ p(n)\), \(l\leq p<\infty\), \(p\neq 2\), \(n\geq 3\). These can be characterized as the norm-convex (in the sense of Menger) subsets of \(\ell^ 2(n)\)-normals having at most two nonzero coordinates. The results obtained by the author complete the results of \textit{L. Hetzelt} [Dissertation, Univ. of Erlangen-Nürnberg (1981) and Acta Math. Hung 45, 53-68 (1985; Zbl 0592.41043)].
scientific article; zbMATH DE number 4075727
scientific article