Note on the Fenchel transform in the Heisenberg group

TL;DR

This paper extends the concept of convexity and Fenchel transform to the Heisenberg group, a fundamental structure in sub-Riemannian geometry, demonstrating that convex functions can be characterized similarly to the Euclidean case.

Contribution

It introduces a new definition of convexity and Fenchel transform adapted to the Heisenberg group, highlighting the role of the H-subdifferential in this setting.

Findings

Convex functions on the Heisenberg group can be characterized via their iterated Fenchel transform.

The properties of the H-subdifferential are crucial for this characterization.

The results parallel classical Euclidean convex analysis in a sub-Riemannian context.

Abstract

Given a real-valued function defined on the Heisenberg group, we provide a definition of abstract convexity and Fenchel transform that takes into account the sub-Riemannian structure of the group. In our main result, we prove that, likewise the Euclidean case, a convex function can be characterized via its iterated Fenchel transform; the properties of the H-subdifferential play a crucial role.