  Given a matrix $A$, a linear feasibility problem (of which linear
classification is a special case) aims to find a solution to a primal problem
$w: A^Tw > \textbf{0}$ or a certificate for the dual problem which is a
probability distribution $p: Ap = \textbf{0}$. Inspired by the continued
importance of "large-margin classifiers" in machine learning, this paper
studies a condition measure of $A$ called its \textit{margin} that determines
the difficulty of both the above problems. To aid geometrical intuition, we
first establish new characterizations of the margin in terms of relevant balls,
cones and hulls. Our second contribution is analytical, where we present
generalizations of Gordan's theorem, and variants of Hoffman's theorems, both
using margins. We end by proving some new results on a classical iterative
scheme, the Perceptron, whose convergence rates famously depends on the margin.
Our results are relevant for a deeper understanding of margin-based learning
and proving convergence rates of iterative schemes, apart from providing a
unifying perspective on this vast topic.
