arXiv:2003.06682v1 [math.OC] 14 Mar 2020 The method of nose stretching in Newton’s least resistance problem for convex bodies Alexander Plakhov ∗ March 17, 2020 Abstract We consider Newton’s problem inf {  Ω (1 + |u(x, y)| 2 ) −1 dxdy : the function u :Ω → R is concave and 0 ≤ u(x, y) ≤ M for all (x, y) ∈ Ω= {(x, y): x 2 +y 2 ≤ 1}} and its generalizations. In the paper[3] it is proved that if a solution u is C 2 in an open set U⊂ Ω then det D 2 u =0 in U . It follows that graph(u)⌋ U does not contain extreme points of the subgraph of u. In this paper we prove a somewhat stronger result. Namely, there exists a solution u possessing the following property. If U⊂ Ω is open and all points of the surface graph(u⌋ U ) are regular, then this surface does not contain extreme points of the convex body C u = {(x,y,z):(x, y) ∈ Ω, 0 ≤ z ≤ u(x, y)}. As a consequence, we have C u = Conv( SingC u ), where SingC u denotes the set of singular points of ∂C u . We prove a similar result for a generalized Newton’s problem. Mathematics subject classifications: 52A15, 52A40, 49Q10 Key words and phrases: Convex bodies, Newton’s problem of min- imal resistance, the method of nose stretching. 1 Introduction and statement of the results Introduce some notation. A convex body C is a compact convex set with nonempty interior. A point ξ ∈ ∂C is called singular if there is more than * Center for R&D in Mathematics and Applications, Department of Mathematics, Uni- versity of Aveiro, Portugal and Institute for Information Transmission Problems, Moscow, Russia. 1