I will talk about sharp isoperimetric inequalities on the hypercube refining several
results due to Kahn--Park and Talagrand.  In the second half of the talk I will present  Talagrand's isoperimetric inequalities
for functions taking values in a Banach space having finite cotype. In addition to that, several different proofs
 of recently resolved Talagrand's conjecture will be presented. 

This is joint with David Beltran and José Ramón Madrid Padilla.