Geodesic
Regularity of convex functions on Heisenberg groups
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 4, pp. 847-868

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We discuss differentiability properties of convex functions on Heisenberg groups. We show that the notions of horizontal convexity (h-convexity) and viscosity convexity (v-convexity) are equivalent and that h-convex functions are locally Lipschitz continuous. Finally we exhibit Weierstrass-type h-convex functions which are nowhere differentiable in the vertical direction on a dense set or on a Cantor set of vertical lines.

Classification : 43A80, 26B25, 49L25 
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     author = {Balogh, Zolt\'an M. and Rickly, Matthieu},
     title = {Regularity of convex functions on {Heisenberg} groups},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {847--868},
     publisher = {Scuola normale superiore},
     volume = {Ser. 5, 2},
     number = {4},
     year = {2003},
     mrnumber = {2040646},
     zbl = {1121.43007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ASNSP_2003_5_2_4_847_0/}
}
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Balogh, Zoltán M.; Rickly, Matthieu. Regularity of convex functions on Heisenberg groups. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 4, pp. 847-868. http://geodesic.mathdoc.fr/item/ASNSP_2003_5_2_4_847_0/