24 nov 2005

**Abstract.**

Dear All,

next Thursday, 24 November, at 17:30 in ``Sala dei Seminari'' of the Department of Mathematics

Aldo Pratelli, from Pavia University will present

``The sharp quantitative estimate for the isoperiometric inequality''

The abstract follows.

The classical isoperimetric inequality states that, given a set E in
R^{n} with the same volume of the unit ball B, the perimeter P(E) of E is
greater than the perimeter P(B) of B. Moreover, if the isoperimetric
deficit D(E)=P(E)-P(B) equals 0, than E coincides with (a translation of)
B. A quantitative version of the isoperimetric inequality consists in
showing that L(E)<D(E)^{p,} where the Fraenkel asymmetry L(E) of E is
defined as the volume of the symmetric difference between E and a suitable
translation of B (the
translation minimizing L(E)!). We will prove the above inequality with
p=1*2, showing also that this is sharp; this result gives a positive
answer to a to a conjecture by Hall (given also, in a weaker version, by
Bonnesen).
*