First we define microscopic sets on the real axis and study their relation to the sets of Hausdorff and Lebesgue measure zero and the sets of first category. In the second part, we prove the Ekeland's variational principle and its equivalence with the the Daneš's drop theorem, the Brézis-Browder's theorem, the Phelps' lemma and the Caristi-Kirks's theorem. Furthermore, we discuss its relation to the Bishop-Phelps' theorem. Doing so we define the notion of a drop as the convex hull of a set and a point. In the third part we prove that the drop property equals reflexivity in some sense. A space has the drop property if it is possible to find the drop from the Daneš's theorem even in a more general case than the theorem itself guarantees. Furthermore, we characterize this property using the approximative compactness. Last, we study the microscopic drop property that is more relaxed than the original drop property. We find out that those two notions are for certain sets in reflexive spaces equivalent.

First we define microscopic sets on the real axis and study their relation to the sets of Hausdorff and Lebesgue measure zero and the sets of first category. In the second part, we prove the Bishop-Phelps' theorem and its equivalence with the Ekeland's variational principle, the Daneš's drop theorem, the Brézis-Browder's theorem and the Caristi-Kirks's theorem. Doing so we define the notion of a drop as the convex hull of a set and a point. In the third part we prove that the drop property equals reflexivity in some sense. A space has the drop property if it is possible to find the drop from the Daneš's theorem even in a more general case than the theorem itself guarantees. Furthermore, we characterize this property using the approximative compactness. Last, we study the microscopic drop property that is more relaxed than the original drop property. We find out that those two notions are for noncompact sets in reflexive spaces equivalent. Powered by TCPDF (www.tcpdf.org)