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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.
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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.
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