Shape optimization problems for 3-dimensional bodies, moving in a planet atmosphere

This paper presents the statements and analytical solutions of optimization problems of finding optimal 3-dimensional body shapes from viewpoint of minimum of radiation heat transfer. In a class of slender bodies possessing homotetic property the initial optimization problem may be reduced to two separated problems of finding optimal longitudinal and transverse contours. From mathematical point of view these problems are variational ones with a glance to various isoperimetric and boundary conditions. The investigation of the problem of determining the optimal transversal contour has shown that there exists a class of variational solutions composed of $n$ identical cycles.