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Under mean radius of curvature flow, a closed convex surface in Euclideanspace is known to expand exponentially to infinity. In the 3-dimensional casewe prove that the oriented normals to the flowing surface converge to theoriented normals of a round sphere whose centre is determined by the initialsurface.  To prove this we show that the oriented normal lines, considered as a surfacein the space of all oriented lines, evolve by a parabolic flow which preservesthe Lagrangian condition. Moreover, this flow converges to a holomorphicLagrangian section, which form the set of oriented lines through a point.  The coordinates of this centre point are projections of the support functioninto the first non-zero eigenspace of the spherical Laplacian and are given byexplicit integrals of initial surface data.

A Converging Lagrangian Curvature Flow in the Space of Oriented Lines