  The lack of an inner product structure in Banach spaces yields the motivation
to introduce a semi-inner product with a more general axiom system, one missing
the requirement for symmetry, unlike the one determing a Hilbert space. We use
it on a finite dimensional real Banach space $(\X, \| \cdot\|)$ to define and
investigate three concepts. First, we generalize that of \emph{antinorms},
already defined in Minkowski planes, for even dimensional spaces. Second, we
introduce \emph{normality maps}, which in turn leads us to the study of
\emph{semi-polarity}, a variant of the notion of polarity, which makes use of
the underlying semi-inner product.
