We derive two sets of inequalities for regular near polygons and study the case where one or more of these inequalities become equalities. This will allow us to obtain two characterization results for dual polar spaces. Our investigation will also have implications for triple intersection numbers and m-ovoids in regular near polygons. In particular, we obtain new results on triple intersection numbers in generalized hexagons of order (s,s^3), s>=2, and prove that no finite generalized hexagon of order (s,s^3), s>=2, can have 1-ovoids. We also show that in one case, the existence of a 1-ovoid would allow a construction of a strongly regular graph srg(47125,12012,3575,2886).