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A parallel flow of non-interacting point particles is incident on a body atrest. When hitting the body’s surface, the particles are reflected elastically.Assuming that each particle hits the body at most once (the single impactcondition (SIC)), the force of resistance of the body along the flow directioncan be written down in a simple analytical form.The problem of minimal resistance within this model was first consideredby Newton (Newton 1687 Philosophiae Naturalis Principia Mathematica) inthe class of bodies with a fixed length M along the flow direction and with afixed maximum orthogonal cross section  , under the additional conditionsthat the body is convex and rotationally symmetric. Here we solve the problem(first stated in Buttazzo et al 1995 Minimum problems over sets of concavefunctions and related questions Math. Nachr. 173 71–89) for the wider classof bodies satisfying the SIC and with the additional conditions removed. Thescheme of solution is inspired by Besicovitch’s method of solving the Kakeyaproblem (Besicovitch 1963 The Kakeya problem Am. Math. Mon. 70 697–706). If   is a disc, the decrease of resistance as compared with the originalNewton problem is more than twofold; the ratio tends to 2 as M 0 and to20.25 as M  . We also prove that the infimum of resistance is 0 for a widerclass of bodies with both single and double reflections allowed

Newton’s problem of minimal resistance under the single-impact assumption