  A Fourier restriction estimate is obtained for a broad class of conic
surfaces by adding a weight to the usual underlying measure. The new
restriction estimate exhibits a certain affine-invariance and implies the sharp
$L^p-L^q$ restriction theorem for compact subsets of a type $k$ conical
surface, up to an endpoint. Furthermore, the chosen weight is shown to be, in
some quantitative sense, optimal. Appended is a discussion of type k conical
restriction theorems which addresses some anomalies present in the existing
literature.
