On the topology of the Steiner symmetrization mapping

We prove that the Steiner symmetrization mapping on the hyperspace of convex bodies in ℝ² is soft and homeomorphic to a fibration in the bundle of Q-manifolds over any compact subset in the hyperspace of symmetric nonpolyhedral subsets. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37, 1994, pp. 9–12.     

On the change in moduli under circular symmetrization

It is shown that under circular symmetrization of multiply-connected domains the moduli of families of curves separating two particular boundary components can be changed in any direction. This is also true for the moduli of families of arcs joining particular boundary components of a domain.Translated from Matematicheskie Zametki, Vol. 10, No. 5, pp. 527–532, November, 1971.     

On a certain kind of symmetrization and its applications

Summary A simple new method of symmetrization is defined and the change of various geometrical and physical quantities under this transformation investigated. To Mauro Picone on his 70^th birthday.     

Isomorphic Steiner symmetrization

This paper proves that there exist 3n Steiner symmetrizations that transform any convex set Kⁿ into an isomorphic Euclidean ball; i.e. if vol(K)=vol(D_n) where D_n is the standard Euclidean unit ball, then K can be transformed into a body K such that c₁D_nKc₂D_n, where c₁,c₂ are numerical constants. Moreover, for any c>2, cn symmetrizations are also enough.     

Compactness via symmetrization

Consider two types of translation-invariant functionals and on , and a sequence of functions f_n whose corresponding symmetric rearrangements are convergent. We show that f_n themselves converge up to translations if either or . These compactness results lead to applications in variational problems and stability problems in stellar dynamics.     

Continuous symmetrization of sets

This article presents a solution to a problem on continuous symmetrization of sets that was originally stated by Polya and Szegö.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 185, pp. 125–139, 1990.     

Homogeneous programming and symmetrization

A polyhedral or piecewise linear homogeneous programming problem is shown through symmetrization to be equivalent to a linear one, yielding a duality theorem for polyhedral homogeneous programming. As a consequence of this duality, it follows that the simplex method can be used to solve such problems.     

Two-point symmetrization and convexity

We prove a conjecture of R. Schneider: the spherical caps are the only spherically convex bodies of the sphere which remain spherically convex after any two-point symmetrization. More generally, we study the relationships between convexity and two-point symmetrization in the Euclidean space and on the sphere. Received: 4 April 2003     

Elliptic equations and Steiner symmetrization

We give some comparison results for elliptic equations by using Steiner symmetrization. © 1996 John Wiley & Sons, Inc.     

On the jets of foliation respecting maps

Using Weil algebra techniques, we determine all finite dimensional homomorphic images of germs of foliation respecting maps.     

Higher-order symmetrization inequalities and applications

We prove new extended forms of the Pólya-Szegö symmetrization principle. As a consequence new sharp embedding theorems for generalized Sobolev and Besov spaces are proved.     

Weighted Dirichlet-type inequalities for Steiner symmetrization

In this paper weighted Dirichlet-type inequalities for Steiner symmetrization are proved. Similar inequalities were known for the so-called starshaped rearrangements. Furthermore it is shown that the Steiner symmetrization is a mapping from into itself. Received April 4, 1997 / Accepted December 5, 1997     

Generalized symmetrization in enveloping algebras

Let g be a Lie algebra,S(g) the symmetric algebra,U(g) the universal enveloping algebra, andZ(g) the center ofU(g). The aim of this paper is to discuss a construction of a class of linear isomorphisms :S(g)U(g) which commute with the adjoint representation. Applications to constructing a basis inZ(g) for classical g are also sketched.Supported by the Russian Foundation for Basic Research under grant 95-01-00814     

Isomorphic Steiner symmetrization of p-convex sets

In this paper we show that given a \(p\) -convex set \(K \subset \mathbb{R }^n\) , there exist \(5n\) Steiner symmetrizations that transform it into an isomorphic Euclidean ball. That is, if \(|K| = |D_n| = \kappa _n\) , we may symmetrize it, using \(5n\) Steiner symmetrizations, into a set \(K'\) such that \(c_p D_n \subset K' \subset C_p D_n\) , where \(c_p\) and \(C_p\) are constants dependent on \(p\) only.     

A symmetrization for finite two-person games

The symmetrization method of Gale, Kuhn and Tucker for matrix games is extended for bimatrix games. It is shown that the equilibria of a bimatrix game and its symmetrization correspond two by two. A similar result is found with respect to quasi-strong, regular and perfect equilibria.