#### 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 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.

## Move your mouse on the pictures to animate

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 , 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.

## Move your mouse on the pictures to animate

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