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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 Ω ⊂ R2 and a piecewise smooth nonpositivefunction u : ¯Ω → R vanishing on ∂Ω. Consider a flow of point particles falling vertically down andreflected elastically from the graph of u. It is assumed that each particle is reflected no more thanonce (no multiple reflections are allowed); then the resistance of the graph to the flow is expressedas R(u; Ω) = 1|Ω|  Ω(1 + |∇u(x)|2)−1dx. We need to find infΩ,u R(u;Ω). One can easily see that|∇u(x)| < 1 for all regular x ∈ Ω, and therefore one always has R(u; Ω) > 1/2. We prove that theinfimum of R is exactly 1/2. This result is somewhat paradoxical, and the proof is inspired by, andpartly similar to, the paradoxical solution given by Besicovitch to the Kakeya problem [Amer. Math.Monthly, 70 (1963), pp. 697–706]

The Problem of Minimal Resistance for Functions and Domains