# Prescribing The Gauss Curvature Of Convex Bodies In Hyperbolic Space

## TL;DR

This paper extends the concept of Gauss curvature measure to convex bodies in hyperbolic space and provides a complete solution, including existence and uniqueness, for Alexandrov's problem in this non-Euclidean setting.

## Contribution

It introduces a new framework for Gauss curvature measure in hyperbolic space and solves Alexandrov's problem with novel methods, establishing existence and uniqueness.

## Key findings

- Complete solution to Alexandrov's problem in hyperbolic space
- Proof of uniqueness of convex bodies with given curvature measure
- Development of new methods for existence and uniqueness proofs

## Abstract

The Gauss curvature measure of a pointed Euclidean convex body is a measure on the unit sphere which extends the notion of Gauss curvature to non-smooth bodies. Alexandrov's problem consists in finding a convex body with given curvature measure. In Euclidean space, A.D. Alexandrov gave a necessary and sufficient condition on the measure for this problem to have a solution. In this paper, we address Alexandrov's problem for convex bodies in the hyperbolic space H m+1. After defining the Gauss curvature measure of an arbitrary hyperbolic convex body, we completely solve Alexandrov's problem in this setting. Contrary to the Euclidean case, we also prove the uniqueness of such a convex body. The methods for proving existence and uniqueness of the solution to this problem are both new.