Heron's problem in an inner product space

An optimization problem attributed to Heron entails minimizing the distance between two fixed points in the plane subject to the constraint that any path between them must include a stop at a given line. While Fermat's theorem from calculus may be applied, the solution ``from The Book" is almost surely a coordinate-free approach from transformation geometry. In this article, we recast Heron's problem in the context of a finite dimensional inner product space. When the constraint is represented by a hyperplane, the problem responds to a natural analogue of the geometric approach in the classical case. We conclude by offering a modest extension of the method for more general constraining subspaces.