Projecting (N−1)-cycles to zero on hyperplanes in RN+1

The projection of a compact oriented submanifold Mn−1⊂Rn+1 on a hyperplane Pn can fail to bound any region in P. We call this “projecting to zero.” Example: The equatorial S1⊂S2⊂R3 projects to zero in any plane containing the x3-axis. Using currents to make this precise, we show a lipschitz (homology) (n−1)-sphere embedded in a compact, strictly convex hypersurface cannot project to zero on n+1 linearly independent hyperplanes in Rn+1. We also show, using examples, that all the hypotheses in this statement are sharp.