Anti-concentration of polynomials: $L^{p}$ balls and symmetric measures

We begin with the observation, based on previous results, that dimension-free lower bounds on the variance of a polynomial under a log-concave measure yield dimension-free small-ball and Fourier decay estimates. Motivated by this, we establish variance bounds for polynomials on log-concave random vectors beyond the classical setting of product measures. First, we consider the family of uniform measures on the $n$-dimensional isotropic $L^{p}$ balls. We show that for a degree-$d$ homogeneous polynomial $f=\sum_{I}a_{I}x^{I}$, with $\sum_{I}a_{I}^{2}=1$, the only obstruction to a dimension-free lower bound on its variance occurs when $p=d$ is an even integer and the coefficients of $f$ are close to those of $\frac{1}{\sqrt{n}}\left\Vert x\right\Vert _{p}^{p}$. Second, we consider general isotropic log-concave measures that are invariant under coordinate permutations and reflections, and determine the minimal variance for quadratic and cubic polynomials. These variance bounds lead to new dimension-free anti-concentration results in both settings, addressing a natural extension of a question posed by Carbery and Wright.