An extremal property of convex hexagons

The following conjecture is discussed: if K is a plane convex figure and T is a triangle of maximal area contained in K, then K is contained in ?5 \sqrt {5} T. It is shown that it suffices to check the conjecture in the case where K is a convex hexagon, but the conjecture is proved only in the case where K is a pentagon. Bibliography: 2 titles.     

Subdivision of a Continuously Distributed Mass

Several theorems on subdivision of a mass continuously distributed on the plane, projective plane, or in space is proved by topological means. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 299, 2003, pp. 228–240.     

Properties of continuous functions on a normed space and its sphere

Let K ⊂ ℝ³ be a convex body of unit volume. It is proved that K contains an affine-regular pentagonal prism of volume 4( 5 - 2?5 ) \mathord

Parallelepipeds circumscribed about a convex body in 3-space

It is proved that each convex body K ⊂ ℝ³ of volume V(K) is contained in a parallelepiped of volume . Bibliography: 4 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 329, 2005, pp. 79–87.     

Some problems of optimizing shell shape and thickness distribution on the basis of a genetic algorithm

We consider problems related to designing axisymmetric shells of minimal weight (mass) and the development of efficient nonlocal optimization methods. The optimization problems under study consist in simultaneous search for the optimal geometry and the shell thickness optimal distribution from the minimal weight condition under strength constraints and additional geometric constraints imposed on the thickness function, the transverse cross-section radii distribution, and the volume enclosed by the shell. Using the method of penalty functions, we reduce the above optimal design problem to a nonconvex minimization problem for the extended Lagrange functional. To find the global optimum, we apply an efficient genetic algorithm. We present the results of numerical solution of the optimal design problem for dome-like shells of revolution under the action of gravity forces. We present some data characterizing the convergence of the method developed here.     

Affine-inscribed and affine-circumscribed polygons and polytopes

Five theorems on polygons and polytopes inscribed in (or circumscribed about) a convex compact set in the plane or space are proved by topological methods. In particular, it is proved that for every interior point O of a convex compact set in ℝ³, there exists a two-dimensional section through O circumscribed about an affine image of a regular octagon. It is also proved that every compact convex set in ℝ³ (except the cases listed below) is circumscribed about an affine image of a cube-octahedron (the convex hull of the midpoints of the edges of a cube). Possible exceptions are provided by the bodies containing a parallelogram P and contained in a cylinder with directrix P. Bibliography: 29 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 231, 1995, pp. 286–298. Translated by B. M. Bekker.     

Two Unsolved Geometric Problems

Two familiar unsolved geometric problems are discussed: on functions on the 2-sphere and on quadrangles inscribed in a smooth planar Jordan curve. A related question on equipartitioning a mass on a plane is also discussed. Bibliography: 6 titles.     

Optimization of flexible beams

We consider the optimal design problem for cantilever beams of variable rigidity loaded at the free end by an arbitrary transverse force. The value of the cantilever free end vertical displacement serves as the optimality criterion, and the distribution of the cantilever thicknesses (cross-sections) is usually used as the design variable. We present results of an asymptotic analysis and a numerical solution of the optimization problem and discuss specific features of the formation of optimal solutions under nonlinear bending.