  The goal of this paper is to attract attention of the reader to a
dimension-free geometric inequality that can be proved using the classical
needle decomposition. This inequality allows us to derive sharp dimension-free
estimates for the distribution of values of polynomials in n-dimensional convex
bodies. Such estimates, in their turn, lead to a surprising result about the
distribution of zeroes of random analytic functions; informally speaking, we
show that for simple families of analytic functions, there exists a "typical"
distribution of zeroes such that the portion of the family occupied by the
functions whose distribution of zeroes deviates from that typical one by some
fixed amount, is about Const exp{-size of the deviation}.
  The paper is essentially self-contained. When choosing the style, we tried to
make it an enjoyable reading for both a senior undergraduate student and an
expert.
  As to the question "What is new in the paper?", we beleive that the answer to
it is a function of two variables, the first being "what is written" and the
second being "who is reading". Since we have no knowledge of the value of the
second variable, we can only give the range of answers with the first variable
fixed. For the targeted audience it will be the standard range [Nothing,
Everything] (with both endpoints included).
