### Approximation of symmetrizations and symmetry of critical points

#### Abstract

We give a sufficient condition in order that a sequence of cap or Steiner symmetrizations or of polarizations approximates some fixed cap or Steiner symmetrization.

This condition is used to obtain the almost sure convergence for random sequences of symmetrization taken in an appropriate set.

The results are applicable to the symmetrization of sets.

An application is given to the study of the symmetry of critical points obtained by minimax methods based on the

Krasnosel'skiĭ genus.

This condition is used to obtain the almost sure convergence for random sequences of symmetrization taken in an appropriate set.

The results are applicable to the symmetrization of sets.

An application is given to the study of the symmetry of critical points obtained by minimax methods based on the

Krasnosel'skiĭ genus.

#### Keywords

Symmetrization; rearrangement; random approximation of symmetrizations; minimax methods; Krasnosiel'skiĭ genus; symmetry of solutions of quasilinear elliptic problems

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