TY - JOUR
T1 - Monotone functions and maps
AU - Basu, Saugata
AU - Gabrielov, Andrei
AU - Vorobjov, Nicolai
PY - 2013
Y1 - 2013
N2 - In [1] we defined semi-monotone sets, as open bounded sets, definable in an o-minimal structure over the reals (e.g., real semialgebraic or subanalytic sets), and having connected intersections with all translated coordinate cones in Rn . In this paper we develop this theory further by defining monotone functions and maps, and studying their fundamental geometric properties. We prove several equivalent conditions for a bounded continuous definable function or map to be monotone. We show that the class of graphs of monotone maps is closed under intersections with affine coordinate subspaces and projections to coordinate subspaces. We prove that the graph of a monotone map is a topologically regular cell. These results generalize and expand the corresponding results obtained in [1] for semi-monotone sets.
AB - In [1] we defined semi-monotone sets, as open bounded sets, definable in an o-minimal structure over the reals (e.g., real semialgebraic or subanalytic sets), and having connected intersections with all translated coordinate cones in Rn . In this paper we develop this theory further by defining monotone functions and maps, and studying their fundamental geometric properties. We prove several equivalent conditions for a bounded continuous definable function or map to be monotone. We show that the class of graphs of monotone maps is closed under intersections with affine coordinate subspaces and projections to coordinate subspaces. We prove that the graph of a monotone map is a topologically regular cell. These results generalize and expand the corresponding results obtained in [1] for semi-monotone sets.
UR - http://www.scopus.com/inward/record.url?scp=84874038851&partnerID=8YFLogxK
UR - http://dx.doi.org/10.1007/s13398-012-0076-4
U2 - 10.1007/s13398-012-0076-4
DO - 10.1007/s13398-012-0076-4
M3 - Article
SN - 1578-7303
VL - 107
SP - 5
EP - 33
JO - Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas
JF - Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas
IS - 1
ER -