# Prescribing the Gauss curvature of hyperbolic convex bodies

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Philippe Castillon

created by raffero on 29 Dec 2020

modified on 07 Jan 2021

14 jan 2021
-- 17:00

Differential Geometry Seminar Torino (online)

**Abstract.**

The Gauss curvature of a convex body can be seen as a measure on the unit sphere (with some properties). For such a measure $\mu$, the Alexandrov problem consists in proving the existence and uniqueness of a convex body whose curvature measure is $\mu$.

In the Euclidean space, this problem was first solved by Alexandrov, and it was observed later that it is equivalent to an optimal transport problem on the sphere. In this talk I will consider Alexandrov problem for convex bodies of the hyperbolic space. After defining the curvature measure, I will explain what are its main properties. If time permits, I will explain how the optimal transport approach of Alexandrov problem leads to a non-linear Kantorovich problem on the sphere, and how to solve it.

Joint work with Jérôme Bertrand.

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