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**Contractive and optimal sets in modular spaces.**
*(English)*
Zbl 1057.46007

The following notions are applied in approximation theory: let \(C\) be a non-empty subset of a Banach space \(X\); there are considered optimal sets, existence sets of best approximation and contractive sets \(C\). In this paper, the Banach space \(X\) is replaced by a modular space \(X_\rho\) generated by a convex modular \(\rho\). The problem of mutual relations between these notions must be reconsidered since modular spaces are more general than Banach spaces, and in general there are two kinds of convergence in \(X_\rho\): \(\rho\)-convergence and \(\|\;\|_\rho\)-convergence, where \(\|\;\|_\rho\) is the norm generated by \(\rho\). This is nontrivial and the need of such investigations is illustrated in the paper by means of the special case of Musielak-Orlicz sequence spaces \(l_\Phi\) or, more generally, by the case of Köthe spaces.

The sequence \(\Phi=(\varphi_n)\) is said to satisfy the condition \((S)\) if all \(\varphi_n\) are twice differentiable in \([0,\infty)\). Among other things, it is shown that if \(\Phi\) satisfies \((S)\), and \(l_\Phi\) is reflexive modularly strictly convex, then for bounded sets \(C\subset l_\Phi\) the following conditions are mutually equivalent: (a) \(C\) is a strictly \((E,\rho)\)-set, (b) \(C\) is a strongly \(\rho\)-contractive set, (c) \(C\) is an intersection of a countable number of half-spaces generated by strongly \(\rho\)-contractive hyperplanes, (d) \(C\) is strongly \(\rho\)-optimal.

The sequence \(\Phi=(\varphi_n)\) is said to satisfy the condition \((S)\) if all \(\varphi_n\) are twice differentiable in \([0,\infty)\). Among other things, it is shown that if \(\Phi\) satisfies \((S)\), and \(l_\Phi\) is reflexive modularly strictly convex, then for bounded sets \(C\subset l_\Phi\) the following conditions are mutually equivalent: (a) \(C\) is a strictly \((E,\rho)\)-set, (b) \(C\) is a strongly \(\rho\)-contractive set, (c) \(C\) is an intersection of a countable number of half-spaces generated by strongly \(\rho\)-contractive hyperplanes, (d) \(C\) is strongly \(\rho\)-optimal.

Reviewer: Julian Musielak (Poznań)

### MSC:

46A80 | Modular spaces |

46B20 | Geometry and structure of normed linear spaces |

46A45 | Sequence spaces (including Köthe sequence spaces) |

46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |