\(k\)-uniform rotundity of Lorentz-Orlicz spaces (Q1353472)

The author considers \(k\)-uniform \((k\geq 2)\) rotundity of Lorentz-Orlicz spaces \(\Lambda_{\Phi,w}(0,\infty)\) and \(\Lambda_{\Phi,w}(0,1)\). In this paper, there are given two long and very interesting and important proofs of properties. 1) The spaces \(\Lambda_{\Phi,w}\) are \(k\)-uniformly convex if and only if they are uniformly convex. 2) The sequence space \(\ell_{\Phi,w}\) is \(k\)-uniformly convex if and only if \(w\) is regular, \(\Phi\in\Delta_2\) for small values and \(\Phi\) is uniformly convex on the interval \((0,\Phi^{-1}(\alpha))\), where \(\alpha= \left[\sum^{k+ 1}_{i=1} w(i)\right]^{-1}\).