Under what conditions on sequence $(a_i)_{i=1}^{\infty}$, where $a_i \in \mathbb R^n$ can we have $$ m^{-1}\sum_{i=1}^{m} \|x_1-a_i\|\|x_2-a_i\| $$ converges as $m \rightarrow \infty$ uniformly for all $x_1$ and $x_2$ in a compact set $\mathcal X$.
For example, if $m^{-1}\sum_{i=1}^{m} \|a_i\|^2$ and $m^{-1}\sum_{i=1}^{m} a_i$ converges as $m \rightarrow \infty$, then $m^{-1}\sum_{i=1}^{m} \|x-a_i\|^2$ converges uniformly for all $x$ in $\mathcal X$. But I cannot derive the similar conditions for the uniform convergence of $m^{-1}\sum_{i=1}^{m} \|x_1-a_i\|\|x_2-a_i\|$. Can anyone give me some ideas. Thanks~