Why do we care about separating hyperplanes?

Question

I understand their use in the calculus of variations/optimization. Was this the original reason for their interest to mathematicians? Why else do we care about them?

Answer 1

I don't know about the history of this topic, but in machine learning, a linear classifier uses a separating hyperplane to determine if an input is of one class or another.

Answer 2

Separating hyperplanes are useful and mathematically interesting because they take something that is relatively hard to express and difficult to reason about—the difference between two disjoint convex sets—into something that is easy to express and reason about. Hyperplanes seem to be a very natural way to cut up Euclidean space and are easy to express in terms of vector spaces too. The hyperplane separation theorem says that some of the fundamental differences between two general arbitrary (but non-intersecting) convex sets can be reduced to the very simple differences between real numbers by finding an appropriate vector for a scalar product.