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