  We discuss a one-parameter family of transformations which changes sets and
functions continuously into their (k,n)-Steiner symmetrizations. Our
construction consists of two stages. First, we employ a continuous
symmetrization introduced by the author in 1990 to transform sets and functions
into their one-dimensional Steiner symmetrization. Some of our proofs in this
stage rely on a simple rearrangement called polarization. In the second stage,
we use an approximation theorem due to Blaschke and Sarvas to give an inductive
definition of the continuous (k,n)-Steiner symmetrization for any 2\leq k \leq
n. This transformation provides us with the desired continuous path, along
which all basic characteristics of sets and functions vary monotonically. The
latter leads to continuous versions of several convolution type inequalities
and Dirichlet's type inequalities as well as to continuous versions of
comparison theorems for solutions of some elliptic and parabolic partial
differential equations.
