Doubling condition at the origin for non-negative positive definite functions

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.

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