  A classic result by Carbery and Wright states that a polynomial of Gaussian
random variables exhibits anti-concentration in the following sense: for any
degree $d$ polynomial $f$, one has the estimate $P( |f(x)| \leq \varepsilon
\cdot E|f(x)| ) \leq O(1) \cdot d \varepsilon^{1/d}$, where the probability is
over $x$ drawn from an isotropic Gaussian distribution. In this note, we give
an elementary proof of this result for the special case when $f$ is a degree
two non-negative polynomial.
