I faced a hard question in kernel methods theory, which I can't answer for about one week. Initially it was formulated in terms of positive valued functions, but it could be reformulated easier:

Let $\{x_1, \dotsc, x_n\}$ and $\{y_1, \dotsc, y_n\}$ be two sets of real positive numbers. Prove that matrix $A$ is positive definite where $A_{i, j} = \min(x_i\cdot y_j, x_j\cdot y_i)$. Equivalently, I want to prove that for arbitrary $a_1, \dotsc, a_n$ the following inequality holds: $$ \sum_{i,j=1}^n a_i a_j \min(x_i\cdot y_j, x_j\cdot y_i)\ge 0. $$

I've run out of ideas on how it could be proved, so any advice is welcome.