On the extremal points of the ball of the Benamou–Brenier energy

Kristian Bredies, Marcello Carioni*, Silvio Fanzon, Francisco Romero

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

10 Citations (Scopus)
22 Downloads (Pure)


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.

Original languageEnglish
Pages (from-to)1436-1452
Number of pages17
JournalBulletin of the London Mathematical Society
Issue number5
Publication statusPublished - Oct 2021
Externally publishedYes


  • 35F05 (primary)
  • 49J45
  • 49N45
  • 52A05


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