Newton's problem of minimal resistance under the single-impact   assumption
  A parallel flow of non-interacting point particles is incident on a body at rest. When hitting the body's surface, the particles are reflected elastically. Assume that each particle hits the body at most once (SIC condition); then the force of resistance of the body along the flow direction can be written down in a simple analytical form.   The problem of minimal resistance within this model was first considered by Newton (1687) in the class of bodies with a fixed length M along the flow direction and with a fixed maximum orthogonal cross section, under the additional conditions that the body is convex and rotationally symmetric. Here we solve the problem (first stated by Ferone, Buttazzo, and Kawohl in 1995) for the wider class of bodies satisfying SIC and with the additional conditions removed. The scheme of solution is inspired by Besicovitch's method of solving the Kakeya problem. If the maximum cross section is a disc, the decrease of resistance as compared with the original Newton problem is more than twofold; the ratio tends to 2 as M goes to 0 and to 81/4 as M goes to infinity. We also prove that the infimum of resistance is 0 for a wider class of bodies with both single and double impacts allowed. 
Alexander Plakhov
math.CA math.DS