The closed curve of length 4π closest to the sphere

Jeff Pickhardt

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The closed curve of length 4π closest to the sphere

Abstract

Kimberling’s tenth unsolved problem asks for optimal parametric equations for a simple closed curve of arclength 4πon the unit sphere minimizing the mean spherical distance from the sphere to the curve. Precisely, for a simple closed curve C ⊂ S2 let d(x,C) = min_q ∈ C dS2 (x,q) and F(C) = 1 4π S2 d(x,C) dA. We prove unconditionally that the minimum is min F= π 6 + √3−2 = 0.2556495832 ..., attained, uniquely up to an isometry of S2, by an explicit C1,1 curve assembled from four semicircles of spherical radius π/6 and two semicircles of great circles. The lower bound comes from the Crofton–Blaschke–Santal´o formula together with a tube estimate, and the extremal attains it because its π/6-neighborhood tiles S2 exactly once: the six tube pieces are, in each of two hemispheres, the two polar caps and the equatorial band of a single axis. For uniqueness we show that equality forces the cut distance to equal π/6 on both sides everywhere, transfer Ghomi–Wenk’s rigidity machinery for the sphere-inspection problem, and then derive a M¨obius relation along the cut locus which forces the geodesic curvature into {0,±√3}. Writing the moving frame in SO(3) then makes each arc act by a fixed matrix and turns the length constraint into a central element of SU(2), so closure becomes an equation for four points of one circle; solving it classifies the candidates exactly, and the single rival is ruled out in closed form. The same argument bounds F below for every length L≥2π, and we determine exactly which L≤4π attain that bound: only L= 2√2 π, where the minimizer is Ghomi–Wenk’s baseball seam, and L= 4π.