Optimization problems in elementary geometry

Abstract

Optimization, a principle of nature and engineering design, in real life problems is normally achieved by using numerical methods. In this article we concentrate on some optimization problems in elementary geometry and Newtonian mechanics. These include Heron’s problem, Fermat’s principle, Brachistochrone problems, Fagano’s problem, geodesics on the surface of a parallelepiped, Fermat/Steiner problem, Kakeya problem and the isoperimetric problem. Some of these are very old and historically famous problems, a few of which are still unresolved. Close connection between Euclidean geometry and Newtonian mechanics is revealed by the methods used to solve some of these problems. Examples are included to show how some problems of analysis or algebra can be solved by using the results of these geometrical optimization problems.