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In this paper we introduce a modification of the Day norm in $c_0\left(\Gamma \right)$ and investigate properties  of this norm.

In this paper we prove that for each $1<p,\tilde{p}<\infty$, the Banach space $(l^{\tilde{p}},{∥·∥}_{\tilde{p}})$ can be equivalently renormed in such a way that  the Banach space $(l^{\tilde{p}},{∥·∥}_{L,\alpha ,\beta ,p,\tilde{p}})$ is LUR and has a diametrically complete set with empty interior. This result extends the Maluta theorem about existence of such a set in $l^2$ with the Day norm. We also show that the Banach space $(l^{\tilde{p}},{∥·∥}_{L,\alpha ,\beta ,p,\tilde{p}})$ has the weak fixed point property for nonexpansive mappings.