Problems of Minimal Resistance and the Kakeya Problem

Here we solve the problem posed by Comte and Lachand-Robert in [SIAM J. Math. Anal., 34 (2002), pp. 101-120]. Take a bounded domain Omega subset of R-2 and a piecewise smooth nonpositive function u : (Omega) over bar -> R vanishing on partial derivative Omega. Consider a flow of point particles falling vertically down and reflected elastically from the graph of u. It is assumed that each particle is reflected no more than once (no multiple reflections are allowed); then the resistance of the graph to the flow is expressed as R(u; Omega) = 1/vertical bar Omega vertical bar integral(Omega)(1 + vertical bar del u(x)|(2))(-1)dx. We need to find inf(Omega, u) R(u; Omega). One can easily see that vertical bar del u(x)vertical bar < 1 for all regular x is an element of Omega, and therefore one always has R(u; Omega) > 1/2. We prove that the infimum of R is exactly 1/2.This result is somewhat paradoxical, and the proof is inspired by, and partly similar to, the paradoxical solution given by Besicovitch to the Kakeya problem [Amer. Math. Monthly, 70 (1963), pp. 697-706].