  We study upper and lower estimates as well as the asymptotic behavior of the
sharp constant $C=C_n(U,V)$ in the doubling-type condition at the origin \[
\frac{1}{|V|}\int_{V}f(x)\,dx\le C\,\frac{1}{|U|}\int_{U}f(x)\,dx, \] where
$U,V\subset \mathbb{R}^{n}$ are $0$-symmetric convex bodies and $f$ is a
non-negative positive definite function.
