A variation on "Hearing the shape of a drum" for polytopes.

Question

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?

Answer 1

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 $\varphi\in C^2(\mathbb S^{d-1},\mathbb R)$ satisfies for all $x\in \mathbb S^{d-1}$, $i$, $j=1,\dots,d$, and some $r>0$ the inequalities $$|\varphi(x)|+\sqrt{2d^3}|D^i\varphi(x)|+2d^3|D^{ij}\varphi(x)|< r, \qquad\qquad (1)$$ where $$D^i\varphi(x)=\left(\frac{\partial \varphi(u/|u|)}{\partial u_i}\right)_{u=x},\quad D^{ij}\varphi(x)=\left(\frac{\partial^2 \varphi(u/|u|)}{\partial u_i\partial u_j}\right)_{u=x},\quad u\in \mathbb R^d,\ x\in \mathbb S^{d-1}.$$ 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 $P_i=P_i(x)$, $x\in \mathbb S^{d-1}$ denote a spherical harmonic of order i.

• There exists a constant $c_0$ such that for all $c\geq c_0$ $$c+P_{n_1}+\dots+P_{n_m} $$ 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 $C\subset \mathbb R^d$ is a compact convex set whose principal radii of curvature exist and srtictly positive and whose support function $\varphi=\sum\limits_{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 $$P_{0}+P_1\dots+P_{n} $$ 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.