Points of monotonicity in Musielak--Orlicz function spaces endowed with the Luxemburg norm (Q1881534)

For a Banach function lattice \(X\) with the cone \(X^+\) of its positive elements, let \(S(X^+)= S(X)\cap X^+\), \(S(X)\) being the unit sphere in \(X\). Let \(L_M\) be a Musielak-Orlicz space with Luxemburg norm and \(E_M\) the subspace of finite elements of \(L_M\). Necessary and sufficient conditions in order that a point \(x\in S(L^+_M)\) be (a) upper monotone, (b) lower monotone, (c) upper locally uniformly monotone, (d) lower locally uniformly monotone are obtained. Let \(e(t)=\sup \{u\geq 0:M(t,u)=0\}\), then each of the properties: (1) \(L_M\) is strictly monotone, (2) \(L_M\) is upper locally uniformly monotone, (3) \(L_M\) is lower locally uniformly monotone, is equivalent to the statement that \(e(t)=0\) \(\mu\)-a.e. and \(M\in\Delta_2\). Similar results hold for \(S(E_M^+)\).