  The goal of this note is to give the unified approach to the solutions of a
class of isoperimetric problems by relating them to the exterior differential
systems studied by R.~Bryant and P.~Griffiths. In this note we list several
classical by now isopereimetric inequalities which can be proved in a unified
way. This unified approach reduces them to the so-called exterior differential
systems studied by Robert Bryant and Phillip Griffiths. To the best of our
knowledge, this is the first article where this connection is used. After
reviewing a list of classical inequalities (log-Sobolev inequality, Beckner's
inequality, Bobkov's functional isoperimetric inequality and several other
inequalities) we use our method to generate new isoperimetric inequalities, in
particular, we found the sharpening of Beckner--Sobolev inequalities with
Gaussian measure.
  Key words: log-Sobolev inequality, Poincar\'e inequality, Bobkov's
inequality, Gaussian isoperimetry, semigroups, maximum principle,
Monge--Amp\`ere equation with drift, exterior differential systems, backwards
heat equation, (B) theorem
