The Problem of Minimal Resistance, Old and New

Since its original formulation by Isaac Newton in 1685, the problem of determining bodies of minimal resistance moving through a fluid has been one of the classical problems in the calculus of variations. Initially posed for cylindrically symmetric bodies, the problem was later extended to general convex shapes, as explored by G. Buttazzo, V. Ferone, and B. Kawohl [Minimum problems over sets of concave functions and related questions, Math. Nachrichten 173 (1995) 71–89] and G. Buttazzo and B. Kawohl [On Newton's problem of minimal resistance, Math. Intelligencer 15 (1993) 7–12]. Since then, this broader formulation has inspired a number of articles dedicated to the study of the geometric and analytical properties of optimal shapes, with particular attention to their structure, regularity, and behavior under various constraints. In this article, we provide a comprehensive overview of the principal results that have been established, highlighting the main theoretical advancements. Furthermore, we introduce some new directions of research, some of which were described by A. Yu. Plakhov in his book Exterior Billiards: Systems with Impacts Outside Bounded Domains [Springer, Berlin (2012)] that offer promising perspectives for future investigation.