# On the extremal points of the ball of the Benamou-Brenier energy

## TL;DR

This paper characterizes the extremal points of the unit ball in the Benamou-Brenier energy space, revealing they are pairs of measures on characteristic curves, and applies this to sparse solutions in inverse problems.

## Contribution

It provides a novel characterization of extremal points in the Benamou-Brenier energy space and applies it to sparse inverse problem solutions.

## Key findings

- Extremal points are pairs of measures on characteristic curves.
- Representation formula for sparse solutions in inverse problems.
- Extension to a coercive generalization of the energy.

## Abstract

In this paper we characterize the extremal points of the unit ball of the Benamou--Brenier energy and of a coercive generalization of it, both subjected to the homogeneous continuity equation constraint. We prove that extremal points consist of pairs of measures concentrated on absolutely continuous curves which are characteristics of the continuity equation. Then, we apply this result to provide a representation formula for sparse solutions of dynamic inverse problems with finite dimensional data and optimal-transport based regularization.