Geodesic
Equicontinuity of Families of Convex and Concave-Convex Operators
Canadian journal of mathematics, Tome 36 (1984) no. 5, pp. 883-898

Voir la notice de l'article provenant de la source Cambridge University Press

J. M. Borwein has given in [1] a practical necessary and sufficient condition for a convex operator to be continuous at some point. Indeed J. M. Borwein has proved in his paper that a convex operator with values in an order topological vector space F (with normal positive cone F+) is continuous at some point if and only if it is bounded from above by a mapping which is continuous at this point. This result extends a previous one by M. Valadier in [16] asserting that a convex operator is continuous at a point whenever it is bounded from above by an element in F on a neighbourhood of the concerned point. Note that Valadier's result is necessary if and only if the topological interior of F+ is nonempty. Obviously both results above are generalizations of the classical one about real-valued convex functions formulated in this context exactly as Valadier's result (see for example [5]). 
Jouak, Mohamed; Thibault, Lionel. Equicontinuity of Families of Convex and Concave-Convex Operators. Canadian journal of mathematics, Tome 36 (1984) no. 5, pp. 883-898. doi: 10.4153/CJM-1984-050-8
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