On the Positive Definiteness of Some Functions Related to the Schoenberg Problem

For a broad class of functions $f\colon[0,+\infty)\to\mathbb{R}$, we prove that the function $f(\rho^{\lambda}(x))$ is positive definite on a nontrivial real linear space $E$ if and only if $0\le\lambda\le \alpha(E,\rho)$. Here $\rho$ is a nonnegative homogeneous function on $E$ such that $\rho(x)\not\equiv 0$ and $\alpha(E,\rho)$ is the Schoenberg constant.