A smooth convex loop with vanishing projections

Solomon (Projecting codimension-two cycles to zero on hyperplanes in RN+1, Topology (2004) X-ref: doi:10.1016/S0040-9383(03)00043-0) has studied the problem of existence of a simple closed curve in R3 whose projections into planes in three linearly independent directions vanish in the sense of currents. He discovered some nonsmooth examples of such curves, and proved that no examples (smooth or otherwise) exist on strictly convex surfaces. We show that smooth (C∞) examples exist, and may be constructed on convex surfaces.