Rotund renormings in spaces of Bochner integrable functions (Q2789233)

The geometry of Köthe-Bochner spaces \(E\left( X\right) \) has been intensively developed during the last decades. The particular case of Lebesgue-Bochner spaces \(L^{p}\left( X\right) \) has often been considered first. Let \(L^{p}=L^{p}\left( T,\Sigma ,\mu \right)\). Recall that \( L^{p}\left( X\right) \) is the Banach space of all (equivalence classes of) Bochner integrable functions \(f:T\rightarrow X\) such that \(\left\| f\left( \cdot \right) \right\| _{X}\in L^{p}\) and it is endowed with the norm \(\left\| f\right\| _{L^{p}\left( X\right) }=\left\| \left\| f\left( \cdot \right) \right\| _{X}\right\| _{L^{p}}\).NEWLINENEWLINEThe authors study basic geometric properties such as rotundity, uniform rotundity in every direction, local uniform rotundity and midpoint local uniform rotundity. It is known that these properties pass from a Banach space \(X\) to the space \(L^{p}\left( X\right) \) when \(1<p<\infty\). Clearly, this is not the case for \(L^{1}\left( X\right)\). The natural question is whether there is an equivalent norm \(\left| \left| \left| \cdot \right| \right| \right| \) on \(L^{1}\left( X\right) \) such that \(\left( L^{1}\left( X\right) ,\left| \left| \left| \cdot \right| \right| \right| \right) \) is rotund whenever \(X\) is. The authors answer this question affirmatively for rotundity and also for all above mentioned properties. The appropriate renorming is the Orlicz-Bochner norm associated to a suitable Orlicz function and the norm of \(X\).NEWLINENEWLINEThe authors also prove that if \(X\) is uniformly rotund, then the norm \( \left| \left| \left| \cdot \right| \right| \right| \) on \(L^{1}\left( X\right) \) is such that its restriction to every reflexive subspace of \(L^{1}\left( X\right) \) is uniformly rotund.