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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.
@article{ASNSP_2003_5_2_4_847_0,
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/}
}
TY - JOUR AU - Balogh, Zoltán M. AU - Rickly, Matthieu TI - Regularity of convex functions on Heisenberg groups JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2003 SP - 847 EP - 868 VL - 2 IS - 4 PB - Scuola normale superiore UR - http://geodesic.mathdoc.fr/item/ASNSP_2003_5_2_4_847_0/ LA - en ID - ASNSP_2003_5_2_4_847_0 ER -
%0 Journal Article %A Balogh, Zoltán M. %A Rickly, Matthieu %T Regularity of convex functions on Heisenberg groups %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2003 %P 847-868 %V 2 %N 4 %I Scuola normale superiore %U http://geodesic.mathdoc.fr/item/ASNSP_2003_5_2_4_847_0/ %G en %F ASNSP_2003_5_2_4_847_0
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/