Let $\varphi:\mathcal S^{d-1}\longrightarrow \mathbb R_{>0}$ be a strictly positive function describing the boundary $\varphi(\mathbf x)\mathbf x,\mathbf x\in\mathbb S^{d-1}$ of a $d-$dimensional closed convex set $C$ with barycenter at the origin. We consider the decomposition $\varphi=\oplus_{\lambda\in \mathbf{Spec}(\Delta)}\varphi_\lambda$ into eigenvectors associated to distinct eigenvalues of the Laplacian. To what extend does the sequence $\parallel \varphi_\lambda\parallel_{\lambda\in\mathbf{Spec}(\Delta)}$ determine $C$? (This is certainly the case for a sphere of radius $\rho$ since then $\varphi=\varphi_0=\rho$.) Are there examples of two non-isometric convex sets (or even two non-isometric polytopes) for which the two corresponding sequences of norms coincide?

Moreover, the convexity of $C$ gives probably constraints for the norms $\parallel \varphi_\lambda\parallel$ since spherical harmonics associated to high eigenvalues wiggle a lot and have thus to be involved with coefficients that are small with respect to $\parallel \varphi_0\parallel$. What are these constraints?