We introduce a new norm (say α-norm) on L(X), the space of all bounded linear operators defined on a normed linear space X. We explore various properties of the α-norm. In addition, we study several equalities and inequalities of the α-norm of operators on 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 α-norm of operators by using the extreme points of the unit ball of the corresponding spaces. Furthermore, we define a geometric constant (say α-index) associated with X and study properties of the α-index. In particular, we obtain the exact value of the α-index for some polyhedral spaces and complex Hilbert space. Finally, we study the α-index of ℓp-sum of normed linear spaces.