Deviation of polynomials from their expectations and isoperimetry

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 Poincaré 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.

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