Consider the case where f is an elementary function of one variable and you know that its slope between any two points is at most k and at least -k. And suppose you want an upper bound on f(0) but you know the values of f(-2) and f(3). Then f(0) <= f(-2) + 2 * k and f(0) <= f(3) + 3 * k. Taking the minimum of these two right-hand sides leads to a formula analogous to the one for U(x) that you're asking about (with the norm || x - x_i || taking the place of |0 - (-2)| and |0 - 3| in our example).