mg.metric geometry - A variation on "Hearing the shape of a drum" for polytopes.

The short answer is that there are no particular constraints on the spectral decomposition of the function $varphi$, as long as a basic convexity condition is satisfied.

Lemma..Assume that $varphiin C^2(mathbb S^{d-1},mathbb R)$ satisfies for all $xin mathbb S^{d-1}$, $i$, $j=1,dots,d$, and some $r>0$ the inequalities
$$|varphi(x)|+sqrt{2d^3}|D^ivarphi(x)|+2d^3|D^{ij}varphi(x)|< r, qquadqquad (1)$$
where
$$D^ivarphi(x)=left(frac{partial varphi(u/|u|)}{partial u_i}right)_{u=x},quad
D^{ij}varphi(x)=left(frac{partial^2 varphi(u/|u|)}{partial u_ipartial u_j}right)_{u=x},quad uin mathbb R^d, xin mathbb S^{d-1}.$$
Then $varphi_r(x)= r+varphi(x)$ is the support function of a compact convex set in $mathbb R^d$.

Condition (1) can be translated into constraints on the harmonic expansion of the support function.

Theorem. Let $P_i=P_i(x)$, $xin mathbb S^{d-1}$ denote a spherical harmonic of order i.

• There exists a constant $c_0$ such that for all $cgeq c_0$
  $$c+P_{n_1}+dots+P_{n_m} $$
  is the support function of a compact convex set in $mathbb R^d$.


• The subset of those compact convex sets whose support functions are finite sums of spherical harmonics is dense (with respect to the Hausdorff metric) in the set of all convex bodies in $mathbb R^d$.


• If $Csubset mathbb R^d$ is a compact convex set whose principal radii of curvature exist and srtictly positive and whose support function
  $varphi=sumlimits_{n=1}^{infty} P_n $
  belongs to the class $C^k(mathbb S^{d-1},mathbb R)$ with $k>frac{d+4}{2}$, then there is
  an $n_0$ such that the partial sum
  $$P_{0}+P_1dots+P_{n} $$
  is the support function of a convex body in $mathbb R^d$ for any $n>n_0$.

Reference. Geometric Applications of Fourier Series and Spherical Harmonics by H. Groemer.