The theory of inequalities has made significant contributions in many areas of mathematics. The purpose of this dissertation is to employ inequalities in studying the geometry of a Banach space.
Motivated by the Hermite-Hadamard inequality, a new family of norms is defined, which is called the p-HH-norm. 
The research outcomes of this thesis make significant contributions in Banach space theory, the theory of means and the theory of inequalities. These contributions including the characterization of inner product spaces via orthogonality; the extension of means of positive
numbers to a vector space setting; and the developments of some important inequalities, namely the Hermite-Hadamard inequality, Ostrowski inequality and Gruss inequality in linear spaces.