Norm inequalities in $${\mathcal {L}}({\mathcal {X}})$$ and a geometric constant

We introduce a new norm (say \(\alpha \)-norm) on \({\mathcal {L}}({\mathcal {X}}),\) the space of all bounded linear operators defined on a normed linear space \({\mathcal {X}}\). We explore various properties of the \(\alpha \)-norm. In addition, we study several equalities and inequalities of the \(\alpha \)-norm of operators on \({\mathcal {X}}.\) As an application, we obtain an upper bound for the numerical radius of product of operators, which improves a well-known upper bound of the numerical radius for sectorial matrices. We present the \(\alpha \)-norm of operators by using the extreme points of the unit ball of the corresponding spaces. Furthermore, we define a geometric constant (say \(\alpha \)-index) associated with \({\mathcal {X}}\) and study properties of the \(\alpha \)-index. In particular, we obtain the exact value of the \(\alpha \)-index for some polyhedral spaces and complex Hilbert space. Finally, we study the \(\alpha \)-index of \(\ell _p\)-sum of normed linear spaces.