Newton's Problem of Minimal Resistance for Bodies Containing a Half-Space

Abstract

We consider Newton's problem of minimal resistance for unbounded bodies in Euclidean space ℝ^d, d ≥ 2. A homogeneous flow of noninteracting particles of velocity v falls onto an immovable body containing a half-space {x : (x, n) < 0} ⊂ ℝ^d, (v, n) < 0. No restriction is imposed on the number of (elastic) collisions of the particles with the body. For any Borel set A ⊂ {v} ⊥ of finite measure, consider the flow of cross-section A: the part of initial flow that consists of particles passing through A.

We construct a sequence of bodies that minimize resistance to the flow of cross-section A, for arbitrary A. This sequence approximates the half-space; any particle collides with any body of the sequence at most twice. The infimum of resistance is always one half of corresponding resistance of the half-space.