Hermite-Hadamard's Inequality and the p-HH-Norm on the Cartesian Product of Two Copies of a Normed Space

The Cartesian product of two copies of a normed space is naturally equipped with\ud the well-known p-norm. In this paper, another notion of norm is introduced, and will be\ud called the p-HH-norm. This norm is an extension of the generalised logarithmic mean and is\ud connected to the p-norm by the Hermite-Hadamard's inequality. The Cartesian product space\ud (with respect to both norms) is complete, when the (original) normed space is. A proof for the\ud completeness of the p-HH-norm via Ostrowski's inequality is provided. This space is embedded\ud as a subspace of the well-known Lebesgue-Bochner function space (as a closed subspace, when\ud the norm is a Banach norm). Consequently, its geometrical properties are inherited from those\ud of Lebesgue-Bochner space. An explicit expression of the superior (inferior) semi-inner product\ud associated to both norms is considered and used to provide alternative proofs for the smoothness\ud and reflexivity of this space