This is an announcement for the paper "Strongly normal cones and the midpoint locally uniform rotundity" by Konstantin Storozhuk. Abstract: We give the method of construction of normal but not strongly normal positive cones in Banach space. Archive classification: math.FA Mathematics Subject Classification: 46B40 Remarks: 5 pages, 3 figures Submitted from: stork@math.nsc.ru The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1112.1196 or http://arXiv.org/abs/1112.1196