  In the first part we study deviation of a polynomial from its mathematical
expectation. This deviation can be estimated from above by Carbery--Wright
inequality, so we investigate estimates of the deviation from below. We obtain
such estimates in two different cases: for Gaussian measures and a polynomial
of an arbitrary degree and for an arbitrary log-concave measure but only for
polynomials of the second degree. In the second part we deals with
isoperimetric inequality and the Poincar\'e inequality for probability measures
on the real line that are images of the uniform distributions on convex compact
sets in $\mathbb{R}^n$ under polynomial mappings.
