A note on the Artstein-Avidan-Milman's generalized Legendre transforms
This provides a theoretical insight for researchers in convex analysis and information geometry, but it is incremental as it builds on prior work.
The paper shows that generalized Legendre transforms on convex functions correspond to ordinary Legendre transforms on affine-deformed functions, linking these transforms to information geometry.
Artstein-Avidan and Milman [Annals of mathematics (2009), (169):661-674] characterized invertible reverse-ordering transforms on the space of lower-semi-continuous extended real-valued convex functions as affine deformations of the ordinary Legendre transform. In this note, we prove that all those generalized Legendre transforms on functions correspond to the ordinary Legendre transform on dually corresponding affine-deformed functions. That is, generalized convex conjugates are convex conjugates of affine-deformed functions. We conclude this note by sketching how this result can be interpreted from the lens of information geometry.