In the arXiv paper "Schoenberg matrices of radial positive definite functions
and Riesz sequences in $L^2(\mathbb{R}^n)$" by L. Golinskii, M. Malamud and L. Oridoroga, Definition 1.1 states
Let $n \in \mathbb{N}$. A real-valued and continuous function $f$ on $\mathbb{R}_+ = [0,\infty)$ is called a radial positive definite function, if for an arbitrary finite set $\{x_1,\ldots,x_m\}$, $x_k \in \mathbb{R}^n$, and $\{\xi_1,\ldots,\xi_m\} \in \mathbb{C}^m$ $$\sum_{k,j = 1}^{n}f(|x_k-x_j|)\xi_k\overline{\xi}_j \ge 0.$$
Note that this definition is equivalent to saying that the matrix $S \in \mathbb{R}^{m \times m}$ defined by $S(k,j) = f(\|x_k-x_j\|)$ is positive semidefinite for any choice of points $x_1,\ldots,x_m \in \mathbb{R}^n$.
Also, Theorem 1.2 states the following:
A function $f \in \Phi_n$, $f(0) = 1$ if and only if there exists a probability measure $\nu$ on $\mathbb{R}_+$ such that $$f(r) = \int_{0}^{\infty}\Omega_n(rt)\nu(dt), ~~~~ r \in \mathbb{R}_+$$ where $$\Omega_n(s) := \Gamma(q+1)\left(\dfrac{2}{s}\right)^qJ_q(s) = \sum_{j = 0}^{\infty}\dfrac{\Gamma(q+1)}{j!\Gamma(j+q+1)}\left(-\dfrac{s^2}{4}\right)^j, ~~~~ q = \dfrac{n}{2}-1,$$ $J_q$ is the Bessel function of the first kind and order $q$.
(In their paper $\Phi_n$ denotes the set of radial positive definite functions on $\mathbb{R}^n$.)
Theorem 1 in "Metric Spaces and Completely Monotone Functions" by I. J. Schoenberg gives the proof of this result.
By applying this theorem for $n = 3$ and $\nu = \delta_1$, you get that $f(r) = \Omega_3(r) = \dfrac{\sin r}{r}$ is a radial positive definite function in $\mathbb{R}^3$. In other words, for any choice of $m$ points $x_1,\ldots,x_m \in \mathbb{R}^3$, the matrix $S \in \mathbb{R}^{m \times m}$ defined by $S(k,j) = \dfrac{\sin(\|x_k-x_j\|)}{\|x_k-x_j\|}$ is positive semidefinite.
It should be clear that if you choose $m$ points $x_1,\ldots,x_m$ in $\mathbb{R}$ or $\mathbb{R}^2$, then $S$ is still guaranteed to be positive semidefinite. However, that is not the case if the points are chosen in $\mathbb{R}^n$ where $n \ge 4$. joriki's answer gives a nice counterexample for $n \ge 5$. Here is an ugly counterexample (found by randomly generating points in MATLAB) with $6$ points in $\mathbb{R}^4$:
$x_1 = (-0.7,0.8,-0.3,-0.4)$
$x_2 = (0.8,-0.8,-0.9,0.7)$
$x_3 = (-0.1,0.0,0.1,-0.1)$
$x_4 = (-0.8,-0.7,-0.1,0.2)$
$x_5 = (0.2,-0.2,0.5,-0.9)$
$x_6 = (0.4,0.8,0.8,0.9)$
You can check that the $6 \times 6$ sinc distance matrix $S$ for these points has an eigenvalue of $\approx -0.0024103$, and thus, $S$ is not positive semi definite.
Best Answer
Let $M := \max_i x_i$. If we define the map $\phi: \{0, \ldots, M\} \to \{0, 1\}^{M+1}$ by $$\phi(x) = (\mathbf{1}\{x \le 0\}, \mathbf{1}\{x \le 1\}, \mathbf{1}\{x \le 2\}, \ldots, \mathbf{1}\{x \le M\}),$$ we have $\min(x_i, x_j) = \langle \phi(x_i), \phi(x_j) \rangle$.
So for any vector $v$, we have $$v^\top A v = \sum_i \sum_j v_i v_j \langle \phi(x_i), \phi(x_j)\rangle = \left\langle \sum_i v_i \phi(x_i), \sum_j v_j \phi(x_j)\right\rangle = \left\|\sum_i v_i \phi(x_i)\right\|^2 \ge 0.$$