Let $x_1, x_2, \dots x_n$ be fixed real numbers. Consider real numbers $v_1, v_2, \dots v_n$ such that $ v_1 + v_2 +\cdots + v_n > 0$. What condition do the points $v_1, v_2, \dots v_n$ need to satisfy, in order for the function $$ f: \mathbb R \mapsto \mathbb R, f(x) = \sum_{i=1}^n v_i \cdot e^{- \frac{1}{2} \cdot (x_i - x)^2}$$ to only have nonnegative values? ( i.e. $f(x) \geq 0$ for all $x \in\mathbb R$ )
We know that if $v_1, v_2, \dots v_n$ are all positive, then $f(x)$ is always positive as well. Is there a weaker condition than this?