Newton’s aerodynamic problem: an overview of recent results and open questions

Newton (1687) posed the problem of finding a convex axisymmetric bodies of the smallest aerodynamic drag. So, we are looking for the optimal curve which is the generatrix of the body. In 1993, a similar problem was formulated in a wider class of convex (not necessarily symmetric) bodies. This task proved to be much more difficult: it is about finding the optimal surface. The talk will provide an overview of recent results and methods used, and besides, we formulate some open questions. А special attention will be paid to the following statement. If all points of an open (in the relative topology) subset of the boundary of some optimal body are regular (that is, $C^1$-smoothness holds), then this set does not contain any extreme points of the body. This statement is a strengthening of a similar result (Brock, Ferone, Kawohl, 1996), formulated for $C^2$-smooth subsets.