Geometry and Topology Seminar: Stability of the Steiner symmetrization of convex sets
16 February 2017 13:00 in CM221
The isoperimetric inequality for Steiner symmetrization of any codimension is investigated and the equality cases are characterized. Moreover, a quantitative version of this inequality is proven for convex sets.
The importance of the Steiner symmetrization relies upon the fact that it acts monotonically
on many geometric and analytic quantities associated with subsets of R^n, e.g. the perimeter.
A characterization of the sets whose perimeter is preserved under
the Steiner symmetrization of codimension 1 was given by Chlebík, Cianchi and Fusco,
see "The perimeter inequality under Steiner symmetrization: cases of equality",
Ann. of Math. 162, 525–555 (2005).
In a this talk I will present some results in the general case of codimension k, with 1 \leq k \leq n-1,
which have been obtained in collaboration with Marco Barchiesi and Nicola Fusco.
We introduce a different approach, based on the regularity properties of the barycenter of the vertical
sections of a set.
The advantage of this approach is twofold. Firstly, we recover and extend the result proved
by Chlebík, Cianchi and Fusco for k = 1 to any codimension, with a new and simpler proof.
Secondly, we are able to obtain a quantitative isoperimetric estimate for convex sets which,
to the best of our knowledge, is the first result of this kind in the framework of Steiner symmetrization.