Orthogonal subspaces of rearrangement invariant spaces (Q1397520)

The paper studies the properties of the two following characteristics of rearrangement-invariant (r.i.) function spaces \(E\) on \([0,1]\): \[ \begin{aligned} \gamma(E)&= \inf \left\{\| 1+x\| _E : x\in E,\;\| x\| _E=1,\;\int_0^1 x(t) \,dt =0\right\},\\ \beta(E)&= \inf\left\{\| x+y\| _E : x,y\in E,\;\| x\| _E\geq 1,\;\| y\| _E\geq 1,\;xy\in L_1,\;\int_0^1 xy\, dt =0\right\}, \end{aligned} \] The authors prove that \(\beta(E)>0\) if and only if \(E\) is isomorphic to \(L_2\). For \(\gamma(E)\) it is proven that \(\gamma(E)>1\) for \(E\) any uniformly convex r.i. space and for \(\Lambda(\varphi)\) a Lorentz space not isomorphic to \(L_1\) or \(L_\infty\). An upper estimate for \(\gamma(L_p)\) is given for all \(p\in (1,\infty)\). Also, it is proven that for every r.i. space \(E\), \(\gamma(E)\leq (1/2)(\sqrt{5}+1)\). Finally, it is proven that the characteristic \(\gamma(E)\) is unstable with respect to equivalent renorming of \(E\), in fact, for all r.i. spaces \(E\) and all \(\varepsilon>0\), there exists an equivalent norm \(\| \cdot\| _F\) on \(E\) so that \(\gamma(F)<1+\varepsilon\). It is stated without elaboration that the characteristic \(\gamma(E)\) is related to the Jung constant and Chebyshev centers.