Moscow--Petersburg seminar on low-dimensional mathematics SPb, PDMI, 08.03.2002, 16:00--17:45, room 311 A.I.Nazarov. Several one-dimensional variational problems. The talk is dedicated to several problems in variational calculus arising from imbedding theorems. All of them have elementary formulations. Nevertheless, some of them were solved only recently (one problem in 2002), and there is no full answer in others.