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 be a convex body in , containing the origin in its interior. Any convex set admits an exterior unit normal vector field , which is uniquely defined almost everywhere. Let be the probability measure on the unit sphere, obtained by rescaling the -dimensional Hausdorff measure. The Gaussian measure of is by definition the pullback of by the Gauss map . More explicitly,
Since contains the origin in its interior, its boundary can be parameterized by a radial map . For every direction in , lies in the intersection of with the ray . We can again pull-back the measure by the map , thus defining a measure on the unit sphere , which we will call Alexandrov measure.
Alexandrov addressed the question of the existence and uniqueness (up to homotethy) of a convex body with prescribed Alexandrov measure , under some conditions on . The relationship between this reconstruction problem and a problem of optimal transport on the unit sphere for the cost 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:
Given a probability measure on the unit sphere, there exists a convex body such that if and only the following optimal transport problem between and for the cost function admits a solution with finite cost:
where the maximum is taken over functions satisfying the relation , and the infimum is taken over transport plans between and .
Based on this observation we solve previous non-standard optimal transportation problem to reconstruct convex bodies from their Gauss measures.