The Reflection Property in Normed Linear Planes with Applications to Generalized Conics

Abstract

**Please note that the full text is embargoed** ABSTRACT: The ancient Greek mathematician Heron was the first to solve the problem of finding the shortest path from point A to point B on one side of the line L, subject to the condition that the path goes from A to L and then to B (figure 1). Figure 1. His solution involved going from A to point R on L and then to B such that the line segments AR and BR make equal angles with L. This is exactly the path a light ray from A to B if L were a mirror. Heron included this proposition in his book Catoptrica, theory of mirrors. See Kline [5], p.168. We use the method of Lagrange multipliers to extend Heron's problem from the Euclidean plane to real normed linear planes where the unit circles of the norm are strictly convex and are continuously differentiable. We also generalize familiar results from Euclidean geometry on the reflection properties of conics. In this introductory section we briefly review some results from Euclidean geometry and some basic properties of a norm which we will use in the following sections.