A new geometric constant to compare p-angular and skew p-angular distances

The $p$-angular distance was first introduced by Maligranda in 2006, while the skew $p$-angular distance was first introduced by Rooin in 2018. In this paper, we shall introduce a new geometric constant named Maligranda-Rooin constant in Banach spaces to compare $p$-angular distance and skew $p$-angular distance. We denote the Maligranda-Rooin constant as $\mathcal{M} \mathcal{R}_p(\mathcal{X})$. First, the upper and lower bounds for the $\mathcal{M} \mathcal{R}_p(\mathcal{X})$ constant is given. Next, it's shown that, a normed linear space is an inner space if and only if $\mathcal{M} \mathcal{R}_p(\mathcal{X})=1$. Moreover, an equivalent form of this new constant is established. By means of the $\mathcal{M} \mathcal{R}_p(\mathcal{X})$ constant, we carry out the quantification of the characterization of uniform nonsquareness. Finally, we study the relationship between the $\mathcal{M} \mathcal{R}_p(\mathcal{X})$ constant, uniform convexity, uniform smooth and normal structure.