  A support vector machine (SVM) is an algorithm that finds a hyperplane which
optimally separates labeled data points in $\mathbb{R}^n$ into positive and
negative classes. The data points on the margin of this separating hyperplane
are called support vectors. We connect the possible configurations of support
vectors to Radon's theorem, which provides guarantees for when a set of points
can be divided into two classes (positive and negative) whose convex hulls
intersect. If the convex hulls of the positive and negative support vectors are
projected onto a separating hyperplane, then the projections intersect if and
only if the hyperplane is optimal. Further, with a particular type of general
position, we show that (a) the projected convex hulls of the support vectors
intersect in exactly one point, (b) the support vectors are stable under
perturbation, (c) there are at most $n+1$ support vectors, and (d) every number
of support vectors from 2 up to $n+1$ is possible. Finally, we perform computer
simulations studying the expected number of support vectors, and their
configurations, for randomly generated data. We observe that as the distance
between classes of points increases for this type of randomly generated data,
configurations with fewer support vectors become more likely.
