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# Sharpness Results and Knapp's Homogeneity Argument

We prove that the $L^2$ restriction theorem, and $L^p \to L^{p'}$, $\frac{1}{p}+\frac{1}{p'}=1$, boundedness of the surface averages imply certain geometric restrictions on the underlying hypersurface. We deduce that these bounds imply that a certain number of principal curvatures do not vanish.