  In this paper we show that the surface measure on the boundary of a convex
body of everywhere positive Gaussian curvature does not admit a Fourier frame.
This answers a question proposed by Lev and provides the first example of a
uniformly distributed measure supported on a set of Lebesgue measure zero that
does not admit a Fourier frame. In contrast, we show that the surface measure
on the boundary of a polytope always admits a Fourier frame.
  We also explore orthogonal bases and frames adopted to sets under
consideration. More precisely, given a compact manifold $M$ without a boundary
and $D \subset M$, we ask whether $L^2(D)$ possesses an orthogonal basis of
eigenfunctions. The non-abelian nature of this problem, in general, puts it
outside the realm of the previously explored questions about the existence of
bases of characters for subsets of locally compact abelian groups.
