Reconstruction of a convex body from its Gaussian curvature measure

in collaboration with Q. Mérigot

We describe below a method to reconstruct a 3D convex body from its Gaussian curvature measure based on a variational characterization based on optimal transport due to [V. Oliker (2007) and J. Bertrand (2010)]. All details can be found in our paper Discrete optimal transport: complexity, geometry and applications.

Let $K\subseteq R^d$ be a convex body in $R^d$, containing the origin in its interior. Any convex set admits an exterior unit normal vector field $ n_K: \partial K\to S^{d-1}$, which is uniquely defined almost everywhere. Let $\sigma$ be the probability measure on the unit sphere, obtained by rescaling the $(d-1)$-dimensional Hausdorff measure. The Gaussian measure $\mathcal{G}_K$ of $K$ is by definition the pullback of $\sigma$ by the Gauss map $n_K$. More explicitly,

\[ \forall B \subseteq \partial K,~ \mathcal{G}_K(B) := \sigma(n_K(B))\]

Since $K$ contains the origin in its interior, its boundary can be parameterized by a radial map $\rho_K: S^{d-1} \to \partial K$. For every direction $u$ in $S^{d-1}$, $\rho_K(u)$ lies in the intersection of $\partial K$ with the ray $\{tu; t \geq 0\}$. We can again pull-back the measure $\mathcal{G}_K$ by the map $\rho_K$, thus defining a measure on the unit sphere $\mathcal{G}_K^0$, which we will call Alexandrov measure.

\[ \forall B \subseteq S^{d-1},~ \mathcal{G}^0_K(B) := \sigma(n_K\circ \rho_K(B)).\]

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Reconstruction of a polytop (on the right) which Gaussian measure (on the left) is supported on three arc of circles

Alexandrov addressed the question of the existence and uniqueness (up to homotethy) of a convex body with prescribed Alexandrov measure $\mu$, under some conditions on $\mu$. The relationship between this reconstruction problem and a problem of optimal transport on the unit sphere for the cost $c(u,v) = -\log(\max(u . v,0))$ has been first remarked by Oliker, and then used by Bertrand to give a direct variational proof of Alexandrov theorem. Bertrand's version of Alexandrov's theorem says the following:

$ \texttt{\bf Theorem : }$ Given a probability measure $\mu$ on the unit sphere, there exists a convex body $K$ such that $\mathcal{G}^0_K = \mu$ if and only the following optimal transport problem between $\sigma$ and $\mu$ for the cost function $c(u,v) := -\log(\max(u.v,0))$ admits a solution with finite cost:

\[ \sup_{\phi,\psi} \int_{S^{d-1}} \phi(u)\, d\sigma - \int_{S^{d-1}}\psi(v)\, d\mu = \inf_{\pi} \int_{S^{d-1} \times S^{d-1}} c(u,v) \\, d \pi(u,v) \]

where the maximum is taken over functions $\phi,\psi$ satisfying the relation $\phi(u) - \psi(v) \leq c(u,v)$, and the infimum is taken over transport plans between $u$ and $v$.
Based on this observation we solve previous non-standard optimal transportation problem to reconstruct convex bodies from their Gauss measures.

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Reconstruction of OTO's polytop (on the right) from its Gaussian measure (on the left) which is uniform plus a singular linear part