Given a real-valued function defined on the Heisenberg group H, we provide a definition of abstract convexity and Fenchel transform in H, that takes into account the sub-Riemannian structure of the group. In our main result, we prove that, likewise the classical case, a convex function can be characterized via its iterated Fenchel transform; the properties of the H-subdifferential play a crucial role.
Calogero, A., Pini, R. (2010). Note on the Fenchel transform in the Heisenberg group. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 368(1), 69-79 [10.1016/j.jmaa.2010.03.001].
Note on the Fenchel transform in the Heisenberg group
CALOGERO, ANDREA GIOVANNI;PINI, RITA
2010
Abstract
Given a real-valued function defined on the Heisenberg group H, we provide a definition of abstract convexity and Fenchel transform in H, that takes into account the sub-Riemannian structure of the group. In our main result, we prove that, likewise the classical case, a convex function can be characterized via its iterated Fenchel transform; the properties of the H-subdifferential play a crucial role.
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