Convex functions in Carnot Groups: old and new

In this talk we consider convex functions on general Carnot groups. We consider several definitions, prove their equivalence, and establish basic regularity properties. A key observation is the realization that convexity depends only on the horizontal distribution and not on the particular base chosen to represent it . This allows us to use potential theoretic representation formulas developed by Bonfiglioli and Lanconelli to approximate convex functions by smooth convex functions. Another new ingredient is Wang's extension to Carnot groups of Bieske's uniqueness result for ∞-harmonic functions in the Heisenberg group.